1. Introduction
Throughout this paper, 
$(R,\mathfrak {m})$ denotes an 
$n$-dimensional, analytically irreducible, Noetherian, local domain.
The Hilbert–Samuel multiplicity of an 
$\mathfrak {m}$-primary ideal 
$\mathfrak {a}\subset R$ is a fundamental invariant of the singularities of 
$\mathfrak {a}$ and satisfies various convexity properties such as Teissier's Minkowski inequality [Reference TeissierTei78]. In this paper, we consider 
$\mathfrak {m}$-filtrations of 
$R$, which generalize the filtrations of 
$R$ given by the powers of a single ideal, and prove various convexity properties for such filtrations. The results have applications to the study of K-stability, volumes of valuations, and problems in commutative algebra.
An 
$\mathfrak {m}$-filtration is a collection 
$\mathfrak {a}_\bullet =(\mathfrak {a}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$ of 
$\mathfrak {m}$-primary ideals of 
$R$ that is decreasing, graded, and left continuous. The latter three conditions mean 
$\mathfrak {a}_\lambda \subset \mathfrak {a}_{\mu }$ when 
$\lambda >\mu$, 
$\mathfrak {a}_{\lambda }\cdot \mathfrak {a}_{\mu }\subset \mathfrak {a}_{\lambda +\mu }$, and 
$\mathfrak {a}_{\lambda }= \mathfrak {a}_{\lambda -\epsilon }$ when 
$0<\epsilon \ll 1$, respectively. The key examples of 
$\mathfrak {m}$-filtrations are as follows.
(1) A trivial example is given by taking powers
$( \mathfrak {b}^{\lceil \lambda \rceil })_{\lambda \in \mathbb {R}_{>0}}$ of a fixed $\mathfrak {m}$-primary ideal $\mathfrak {b}\subset R$.(2) An important example in this paper is
$\mathfrak {a}_{\bullet }(v):= (\mathfrak {a}_{\lambda }(v))_{\lambda \in \mathbb {R}_{>0}}$, where $v\colon {\rm Frac}(R)^\times \to \mathbb {R}$ is a valuation centered at $\mathfrak {m}$ and $\mathfrak {a}_{\lambda }(v) := \{f \in R\mid v(f) \geq \lambda \}$ (see § 2.2).(3) If
$(\mathfrak {b}_{\lambda })_{\lambda \in \mathbb {Z}_{>0}}$ is a decreasing, graded sequence of $\mathfrak {m}$-primary ideals,Footnote 1 then $(\mathfrak {b}_{\lceil \lambda \rceil })_{\lambda \in \mathbb {R}_{>0}}$ is an $\mathfrak {m}$-filtration. Such sequences have been well studied in the literature [Reference LazarsfeldLaz04, § 2.4.B].
Following work of Ein, Lazarsfeld, and Smith, the multiplicity of an 
$\mathfrak {m}$-filtration is
\[ \mathrm{vol}(\mathfrak{a}_\bullet) := \lim_{ \mathbb{N} \ni m\to\infty} \frac{\ell(R/\mathfrak{a}_m)}{m^n/n!} = \lim_{ \mathbb{N} \ni m\to\infty} \frac{\mathrm{e}(\mathfrak{a}_m)}{m^n} , \]
where the existence of the above limit and the equality were proven in increasing generality by [Reference Ein, Lazarsfeld and SmithELS03, Reference MustaMus02, Reference Lazarsfeld and MustaLM09, Reference CutkoskyCut13, Reference CutkoskyCut14]. This invariant is the local counterpart of the volume of a graded linear series of a line bundle and has been studied both in the context of commutative algebra [Reference Cutkosky, Sarkar and SrinivasanCSS19, Reference CutkoskyCut21] and recently in work of C. Li and others on the normalized volume of a valuation [Reference LiLi18, Reference Li, Liu and XuLLX20].
In this paper, we prove two properties of the multiplicity of 
$\mathfrak {m}$-filtrations. The first is a convexity result, which has applications to volumes of valuations and C. Li's normalized volume function. The second is a generalization of a classical theorem of Rees [Reference ReesRee61] from the setting of ideals to filtrations and results in a characterization of when the Minkowski inequality for filtrations is an equality.
1.1 Multiplicity and geodesics
Given two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$, we define a segment of 
$\mathfrak {m}$-filtrations 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ interpolating between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ by setting
\[ \mathfrak{a}_{\lambda,t} := \sum_{\lambda=(1-t)\mu+t\nu} \mathfrak{a}_{\mu,0} \cap \mathfrak{a}_{\nu,1} . \]
We call 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ the geodesic between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$, since it is the local analogue of the geodesic between two filtrations of the section ring of a polarized variety [Reference Blum, Liu, Xu and ZhuangBLXZ23, Reference RebouletReb22]. This definition is also related to a construction in [Reference Xu and ZhuangXZ21].
In [Reference Blum, Liu, Xu and ZhuangBLXZ23], it was shown that several non-Archimedean functionals from the theory of K-stability are strictly convex along geodesics in the global setting. In a similar spirit, we prove a convexity result in the local setting for the multiplicity along geodesics.
Theorem 1.1 Assume 
$R$ contains a field. If 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ are 
$\mathfrak {m}$-filtrations with positive multiplicity, then the function 
$E(t) \colon [0,1]\to \mathbb {R}$ defined by 
$E(t) := \mathrm {e}(\mathfrak {a}_{\bullet,t})$ satisfies the following properties:
(1)
$E(t)$ is smooth;(2)
$E(t)^{-1/n}$ is concave, meaning
\[ E(t)^{-1/n}\ge (1-t)E(0)^{-1/n}+tE(1)^{-1/n} \quad \text{for all } t\in [0,1]; \]
(3)
$E(t)^{-1/n}$ is linear if and only if $\widetilde {\mathfrak {a}}_{\bullet,0} =\widetilde {\mathfrak {a}}_{c\bullet,1}$ for some $c\in \mathbb {R}_{>0}$.
The term ‘smooth’ in Theorem 1.1(1) means 
$E(t)$ extends to a 
$C^\infty$ function on 
$(-\epsilon,1+\epsilon )$ for some 
$\epsilon >0$. The symbol 
$\widetilde {\cdot }$ in Theorem 1.1(3) denotes the saturation of an 
$\mathfrak {m}$-filtration, which is defined in § 3.1. This notion is an analogue of the integral closure of an ideal in the setting of filtrations and discussed further in § 1.2 below.
The proof of Theorem 1.1 is inspired by a related argument in the global setting [Reference Blum, Liu, Xu and ZhuangBLXZ23] and relies on constructing a measure on 
$\mathbb {R}^2$ that encodes the multiplicities of the filtrations along the geodesic. In the special case when 
$\mathfrak {a}_{\bullet,0}$ is the 
$\mathfrak {m}$-filtration of a valuation minimizing the normalized volume function over a Kawamata log terminal (klt) singularity, the proof of [Reference Xu and ZhuangXZ21] can be used to show Theorem 1.1(2). Theorem 1.1 removes these strong restrictions and is proven without the theory of K-stability for valuations introduced in [Reference Xu and ZhuangXZ21, § 3.1].
1.1.1 Applications to volume
As first defined by Ein, Lazarsfeld, Smith, the volume of a valuation 
$v\colon {\rm Frac}(R)^{\times } \to \mathbb {R}$ centered at 
$\mathfrak {m}$ is
\[ \mathrm{vol}(v) := \mathrm{e}(\mathfrak{a}_\bullet(v)) = \lim_{m \to \infty} \frac{ \ell \big(R/ \mathfrak{a}_{m}(v)\big) }{m^n/n!} . \]
This invariant is a local analogue of the volume of a line bundle and also plays a role in the study of K-stability of Fano varieties and Fano cone singularities.
Theorem 1.1 can be applied to show that the volume of a valuation is strictly log convex on simplices of quasi-monomial valuations in the valuation space of 
$(R,\mathfrak {m})$. This gives an affirmative answer to [Reference Li, Liu and XuLLX20, Question 6.23]. We note that the volume was previously shown to be Lipschitz continuous on such a simplex [Reference Boucksom, Favre and JonssonBFJ14, Corollary D].
Corollary 1.2 (Convexity of volume)
Assume 
$R$ contains a field. Let 
$\eta \in (Y,D_1+\cdots +D_r)$ be a log smooth birational model of 
$(R,\mathfrak {m})$. For any 
$\boldsymbol {\alpha },\boldsymbol {\beta } \in \mathbb {R}_{>0}^{r}$ and 
$t\in (0,1)$,
\[ \mathrm{vol}(v_{(1-t)\boldsymbol {\alpha}+ t \boldsymbol {\beta}})^{-1/n} \geq (1-t)\,\mathrm{vol}(v_{\boldsymbol {\alpha}})^{-1/n} + t \,\mathrm{vol}(v_{\boldsymbol {\beta}})^{-1/n} \]
and equality holds if and only if 
$\boldsymbol {\alpha } = c \boldsymbol {\beta }$ for some 
$c\in \mathbb {R}_{>0}$.
In the above theorem, 
$v_{\boldsymbol {\alpha }}$ denotes the quasi-monomial valuation of 
${\rm Frac}(R)$ with weights 
$\boldsymbol {\alpha }$ on 
$D_1,\ldots, D_r$. See § 2.2.3 for a detailed definition.
1.1.2 Applications to normalized volume
In [Reference LiLi18], Chi Li defined the normalized volume of a valuation over a klt singularity and proposed the problem of studying its minimizer. The notion plays an important role in the study of K-stability of Fano varieties and in the study of klt singularities. The invariant has been extensively studied in the recent years; see [Reference Li, Liu and XuLLX20] and [Reference ZhuangZhu23] for surveys on this topic.
The fundamental problem in the study of the normalized volume function is the stable degeneration conjecture proposed by Li [Reference LiLi18, Conjecture 7.1] and Li and Xu [Reference Li and XuLX18, Conjecture 1.2]. The conjecture predicts that there exists a valuation minimizing the normalized volume function and that the minimizer is unique up to scaling, quasi-monomial, has finitely generated associated graded ring, and induces a degeneration of the klt singularity to a K-semistable Fano cone singularity. These five statements were proven in [Reference BlumBlu18, Reference Li and XuLX18, Reference XuXu20, Reference Xu and ZhuangXZ21, Reference Xu and ZhuangXZ22]. In particular, the stable degeneration conjecture is now a theorem.
Using Theorem 1.1, we revisit Xu and Zhuang's theorem stating that the minimizer of the normalized volume function is unique up to scaling [Reference Xu and ZhuangXZ21]. (The uniqueness was also previously proven in [Reference Li and XuLX18] under the assumption that the minimizer has finitely generated associated ring. The latter was recently shown in [Reference Xu and ZhuangXZ22].) In particular, we give a proof of the uniqueness result independent of the theory of K-semistability for valuations developed in [Reference Xu and ZhuangXZ21].
Corollary 1.3 (Uniqueness of minimizer)
If 
$x\in (X,D)$ is a klt singularity defined over an algebraically closed field of characteristic 
$0$, then any minimizer of 
$\widehat {\mathrm {vol}}_{X,D,x}$ (see Definition 2.8) is unique up to scaling.
The new proof of Corollary 1.3 takes the following direct approach. Fix two valuations 
$v_0$ and 
$v_1$ that minimize 
$\widehat {\mathrm {vol}}_{X,D,x}$ and consider the geodesic 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ between 
$\mathfrak {a}_{\bullet }(v_0)$ and 
$\mathfrak {a}_{\bullet }(v_1)$. Using Theorem 1.1(2), a characterization of the infimum of the normalized volume function in terms of normalized multiplicities [Reference LiuLiu18, Theorem 27], and an inequality of log canonical thresholds [Reference Xu and ZhuangXZ21, Theorem 3.11], we show that 
$\mathrm {e}(\mathfrak {a}_{\bullet,t})$ is linear. Theorem 1.1(3) then implies 
$cv_0=v_1$ for some 
$c>0$.
1.2 Rees's theorem
A theorem of Rees [Reference ReesRee61] states that if 
$\mathfrak {a}\subset \mathfrak {b}$ are two 
$\mathfrak {m}$-primary ideals, then the following statements are equivalent.
(1)
$\mathrm {e}(\mathfrak {a})=\mathrm {e}(\mathfrak {b})$.(2)
$\overline { \bigoplus _{m\in \mathbb {N}} \mathfrak {a}^m}=\overline { \bigoplus _{m \in \mathbb {N}} \mathfrak {b}^m}$.(3)
$\overline {\mathfrak {a}}=\overline {\mathfrak {b}}$.
The symbol 
$\overline {\, \cdot \, }$ in (2) denotes the algebraic closure in 
$R[t]$, while in (3) it denotes the integral closure of an ideal. The equivalence between (2) and (3) follows from definitions.
It is natural to ask for a generalization of the above result for 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet$. In [Reference CutkoskyCut21], Cutkosky studies whether the two conditions
(1)
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\mathfrak {b}_\bullet )$,(2)
$\overline {\bigoplus _{m\in \mathbb {N}} \mathfrak {a}_m}=\overline { \bigoplus _{m \in \mathbb {N}} \mathfrak {b}_m}$
are equivalent. While (2) 
$\Rightarrow$ (1) holds by [Reference Cutkosky, Sarkar and SrinivasanCSS19, Theorem 6.9], (1) 
$\Rightarrow$ (2) can fail even in very simple examples (see Example 3.7). That said, (1) 
$\Rightarrow$ (2) holds for special classes of 
$\mathfrak {m}$-filtrations [Reference CutkoskyCut21, Theorem 1.4].
To remedy this issue, we introduce the saturation 
$\widetilde {\mathfrak {a}}_\bullet$ of an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ in § 3. (The definition may be viewed as a local analogue of a construction studied by Boucksom and Jonsson in [Reference Boucksom and JonssonBJ21]; see § 6.) The saturation is defined using divisorial valuations, analogous to the valuative definition of the integral closure of an ideal. Using this notion, we prove a version of Rees's theorem for filtrations.
Theorem 1.4 (Rees's theorem)
For 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet$, 
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\mathfrak {b}_\bullet )$ if and only if 
$\widetilde {\mathfrak {a}}_\bullet =\widetilde {\mathfrak {b}}_\bullet$.
The above result can be explained as follows. The multiplicity of an 
$\mathfrak {m}$-filtration is determined by its valuative properties, not by the integral properties of its Rees algebra. These two properties coincide for ideals, but do not always coincide for filtrations by Example 3.7.
1.3 Minkowski inequality
By work of Teissier [Reference TeissierTei78], Rees and Sharp [Reference Rees and SharpRS78], and Katz [Reference KatzKat88] in increasing generality, for two 
$\mathfrak {m}$-primary ideals 
$\mathfrak {a}$ and 
$\mathfrak {b}$ of 
$R$,
\[ \mathrm{e}(\mathfrak{a}\mathfrak{b})^{1/n}\leq \mathrm{e}(\mathfrak{a})^{1/n}+\mathrm{e}(\mathfrak{b})^{1/n} \]
and the equality holds if and only if there exist 
$c,d\in \mathbb {Z}_{>0}$ such that 
$\overline {\mathfrak {a}^c}=\overline {\mathfrak {b}^d}$.
For two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$, we let 
$\mathfrak {a}_\bullet \mathfrak {b}_\bullet$ denote the 
$\mathfrak {m}$-filtration 
$( \mathfrak {a}_\lambda \mathfrak {b}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$. Using the saturation of a filtration, we characterize when the Minkowski inequality for filtrations is an equality.
Corollary 1.5 (Minkowski inequality)
For 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ with positive multiplicity,
\begin{equation} \mathrm{e}(\mathfrak{a}_\bullet\mathfrak{b}_\bullet)^{1/n}\leq \mathrm{e}(\mathfrak{a}_\bullet)^{1/n}+\mathrm{e}(\mathfrak{b}_\bullet)^{1/n} \end{equation}
and the equality holds if and only if 
$\widetilde {\mathfrak {a}}_{\bullet } = \widetilde {\mathfrak {b}}_{c\bullet }$ for some 
$c\in \mathbb {R}_{>0}$.
The inequality statement of the above corollary is not new and is due to Musta [Reference MustaMus02], Kaveh and Khovanskii [Reference Kaveh and KhovanskiiKK14], and Cutkosky [Reference CutkoskyCut15] in increasing levels of generality. In the equality statement, the forward implication follows easily from [Reference CutkoskyCut21, Theorem 10.3] and the definition of the saturation, while the reverse implication relies on Theorem 1.4.
Remark 1.6 (Relation to work of Cutkosky)
Cutkosky proved a version of the equality part of Corollary 1.5 in the special case when 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ are bounded filtrations [Reference CutkoskyCut21, Definition 1.3], which roughly means that their integral closure is induced by a finite collection of divisorial valuations [Reference CutkoskyCut21, Theorem 1.6]. For such filtrations, the integral closure and saturation agree by Lemma 3.20. Thus, Corollary 1.5 may be viewed as a generalization of the latter result. Similarly, Theorem 1.4 may be viewed as a generalization of [Reference CutkoskyCut21, Theorem 1.4].
This paper is organized as follows: In § 2, we recall definitions and basic facts concerning filtrations, valuations, and multiplicities. In § 3, we introduce the saturation of a filtration and prove Theorem 1.4. In § 4, we define the geodesic between two filtrations and prove Theorem 1.1. In § 5, we deduce Corollaries 1.2, 1.3, and 1.5 as consequences of results in the previous two sections. In § 6, we discuss relations between the results in this paper and global results in the K-stability literature. The appendix of the paper is devoted to an alternate proof of a special case of Theorem 1.1 using the theory of Okounkov bodies.
2. Preliminaries
Throughout this section, 
$(R,\mathfrak {m},\kappa )$ denotes an 
$n$-dimensional, analytically irreducible,Footnote 2 Noetherian, local domain. We set 
$X:= \mathrm {Spec}(R)$ and write 
$x\in X$ for the closed point corresponding to 
$\mathfrak {m}$.
2.1 Filtrations
Definition 2.1 An 
$\mathfrak {m}$-filtration is a collection 
$\mathfrak {a}_\bullet =(\mathfrak {a}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$ of 
$\mathfrak {m}$-primary ideals of 
$R$ such that
(1)
$\mathfrak {a}_\lambda \subset \mathfrak {a}_{\mu }$ when $\lambda >\mu$,(2)
$\mathfrak {a}_{\lambda } = \mathfrak {a}_{\lambda -\epsilon }$ when $0<\epsilon \ll 1$, and(3)
$\mathfrak {a}_{\lambda } \cdot \mathfrak {a}_{\mu } \subset \mathfrak {a}_{\lambda + \mu }$ for any $\lambda,\mu \in \mathbb {R}_{>0}$.
By convention, we set 
$\mathfrak {a}_{0}:= R$. The definition is a local analogue of a filtration of the section ring of a polarized variety in [Reference Boucksom, Hisamoto and JonssonBHJ17].
For 
$\lambda \in \mathbb {R}_{\ge 0}$, define 
$\mathfrak {a}_{>\lambda }:=\bigcup _{\mu >\lambda }\mathfrak {a}_\mu$. If 
$\lambda$ satisfies 
$\mathfrak {a}_{>\lambda }\subsetneq \mathfrak {a}_\lambda$, then we say 
$\lambda$ is a jumping number of the filtration.
The scaling of an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ by 
$c\in \mathbb {R}_{>0}$ is 
$\mathfrak {a}_{c\bullet }:=(\mathfrak {a}_{c \lambda } )_{\lambda \in \mathbb {R}_{>0}}$. The product of two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ is 
$\mathfrak {a}_\bullet \mathfrak {b}_\bullet :=(\mathfrak {a}_\lambda \mathfrak {b}_\lambda )_{\lambda \in \mathbb {R}_{>0}}$. Both are again 
$\mathfrak {m}$-filtrations.
Definition 2.2 An 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ is linearly bounded if there exists a constant 
$c>0$ such that 
$\mathfrak {a}_{\lambda } \subset \mathfrak {m}^{\lceil c \lambda \rceil }$ for all 
$\lambda \in \mathbb {R}_{>0}$.
Lemma 2.3 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. If 
$\mathfrak {a}_\bullet$ is linearly bounded, then there exists 
$c\in \mathbb {R}_{>0}$ such that 
$\mathfrak {a}_{c\lambda } \subset \mathfrak {b}_\lambda$ for all real numbers 
$\lambda \geq 1$
Proof. Since 
$\mathfrak {a}_\bullet$ is linearly bounded, there exists 
$c_0 \in \mathbb {R}_{>0}$ such that 
$\mathfrak {a}_{\lambda } \subset \mathfrak {m}^{ \lceil c_0\lambda \rceil }$ for all 
$\lambda >0$. Since 
$\mathfrak {b}_1$ is an 
$\mathfrak {m}$-primary ideal, there exists 
$d\in \mathbb {Z}_{>0}$ such that 
$\mathfrak {m}^{d} \subset \mathfrak {b}_1$. If we set 
$c:= 2d/c_0$, then
\[ \mathfrak{a}_{ c\lambda } \subset \mathfrak{m}^{\lceil 2d \lambda \rceil } \subset \mathfrak{m}^{d \lceil \lambda \rceil } \subset (\mathfrak{b}_1)^{\lceil \lambda \rceil} \subset \mathfrak{b}_{\lceil \lambda \rceil } \subset \mathfrak{b}_{\lambda} \]
for all 
$\lambda \geq 1$. (The second inclusion uses that 
$d\lceil \lambda \rceil \leq d(\lambda +1) \leq 2d \lambda \leq \lceil 2 d \lambda \rceil$).
2.2 Valuations
Let 
$(\Gamma,\ge )$ be a totally ordered abelian group. A 
$\Gamma$-valuation of 
$R$ is a map 
$v\colon {\rm Frac(R)}^\times \to \Gamma$ such that
(1)
$v(fg)=v(f)+v(g)$, and(2)
$v(f+g)\ge \min \{v(f),v(g)\}$.
By convention, we set 
$v(0):= \infty$. We say 
$v$ is centered at 
$\mathfrak {m}$ if 
$v\geq 0$ on 
$R$ and 
$v>0$ on 
$\mathfrak {m}\subset R$.
A valuation 
$v$ of 
$R$ induces a valuation ring 
$R_{v} : = \{ f \in K \mid v(f) \geq 0\}$. We write 
$\mathfrak {m}_v$ for the maximal ideal of 
$R_v$ and 
$\kappa _{v}:=R_v/\mathfrak {m}_v$.
2.2.1 Real valuations
When 
$\Gamma =\mathbb {R}$ with the usual order, we say that 
$v$ is a real valuation. We denote by 
$\mathrm {Val}_{R,\mathfrak {m}}$ the set of real valuations centered at 
$\mathfrak {m}$.Footnote 3 In geometric settings, we will instead denote the set by 
$\mathrm {Val}_{X,x}$.
For 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ and 
$\lambda \in \mathbb {R}_{>0}$, we define the valuation ideal
\[ \mathfrak{a}_{\lambda}(v) := \{ f \in R \mid v(f) \geq \lambda \} \]
for each 
$\lambda \in R$. Using (1) and (2), one can show 
$\mathfrak {a}_\bullet (v)$ is an 
$\mathfrak {m}$-filtration.
For 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ and an ideal 
$\mathfrak {a}\subset R$, set 
$v(\mathfrak {a}):= \min \{v(f)\mid f\in \mathfrak {a}\}$. For an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$, set
\[ v(\mathfrak{a}_\bullet):= \lim_{ \mathbb{N}\in m \to \infty} \frac{ v(\mathfrak{a}_m)}{m} = \inf_{m\in \mathbb{N}} \frac{v(\mathfrak{a}_m)}{m} , \]
where the existence of the limit and second equality is [Reference Jonsson and MustaţăJM12, Proposition 2.3].
2.2.2 Divisorial valuations
A valuation 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ is divisorial if
\[ {\rm tr.deg}_{\kappa}(\kappa_v) =n-1. \]
We write 
$\mathrm {DivVal}_{R,\mathfrak {m}}\subset \mathrm {Val}_{R,\mathfrak {m}}$ for the set of such valuations.
Divisorial valuations appear geometrically. If 
$\mu :Y\to X$ is a proper birational morphism with 
$Y$ normal and 
$E\subset Y$ a prime divisor, then there is an induced valuation 
$\mathrm {ord}_{E} \colon {\rm Frac}(R)^\times \to \mathbb {Z}$. If 
$\mu (E) =x$ and 
$c\in \mathbb {R}_{>0}$, then 
$c\cdot \mathrm {ord}_E \in \mathrm {DivVal}_{R,\mathfrak {m}}$. When 
$R$ is excellent, all divisorial valuations are of this form; see, for example, [Reference Cutkosky and SarkarCS22, Lemma 6.5].
2.2.3 Quasi-monomial valuations
In the following construction, we always assume 
$R$ contains a field. Let 
$\mu \colon Y:= \mathrm {Spec}(S) \to X=\mathrm {Spec}(R)$ be a birational morphism with 
$R\to S$ finite type and 
$\eta \in Y$ a not necessarily closed point such that 
$\mathcal {O}_{Y,\eta }$ is regular and 
$\mu (\eta ) =x$. Given a regular system of parameters 
$y_1,\ldots, y_r$ of 
$\mathcal {O}_{Y,\eta }$ and 
$\boldsymbol {\alpha }=(\alpha _1,\ldots, \alpha _r)\in \mathbb {R}_{\geq 0}^r\setminus {\bf 0}$, we define a valuation 
$v_{\bf \alpha }$ as follows. For 
$0\neq f\in \mathcal {O}_{Y,\eta }$, we can write 
$f$ in 
$\widehat {\mathcal {O}}_{Y,\eta } \simeq k(\eta ) [[y_1,\ldots, y_r]]$ as 
$\sum _{\boldsymbol {\beta }\in \mathbb {Z}_{\geq 0}^r} c_{\boldsymbol {\beta }} {y}^{\boldsymbol {\beta }}$ and set
\[ v_{\boldsymbol{\alpha}}(f):= \min \{ \langle \boldsymbol {\alpha}, {\boldsymbol{\beta}} \rangle \mid c_{\boldsymbol{\beta}} \neq 0 \}. \]
A valuation of the above form is called quasi-monomial.
Let 
$D= D_{1}+\cdots +D_r$ be a reduced divisor on 
$Y$ such that 
$y_i=0$ locally defines 
$D_i$ and 
$\mu (D_i)= x$ for each 
$i$. We call 
$\eta \in (Y,D)$ a log smooth birational model of 
$X$. We write 
$\mathrm {QM}_\eta (Y,E) \subset \mathrm {Val}_{X,x}$ for the set of quasi-monomial valuations that can be described at 
$\eta$ with respect to 
$y_1,\ldots, y_r$ and note that 
$\mathrm {QM}_\eta (Y,D) \simeq \mathbb {R}^r_{\geq 0} \setminus \boldsymbol {0}$.
2.2.4 Izumi's inequality
The order function 
$R\setminus 0 \to \mathbb {N}$ is defined by
\[ \mathrm{ord}_{\mathfrak{m}}(f):= \max \{ k \in \mathbb{Z}_{\geq 0} \mid f\in \mathfrak{m}^k \}. \]
The following version of Izumi's inequality compares 
$\mathrm {ord}_{\mathfrak {m}}$ to a fixed quasi-monomial valuation.
Lemma 2.4 Let 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$. If (i) 
$v$ is divisorial or (ii) 
$R$ contains a field and 
$v$ is quasi-monomial, then there exists a constant 
$c>0$ such that
\[ v(\mathfrak{m}) \cdot \mathrm{ord}_\mathfrak{m}(f)\leq v(f)\leq c\cdot\mathrm{ord}_\mathfrak{m}(f) \]
for all 
$f\in R$. In particular, 
$\mathfrak {a}_\bullet (v)$ is linearly bounded.
Proof. Since 
$f \in \mathfrak {m}^{\mathrm {ord}_{\mathfrak {m}}(f)}$ by definition, 
$v(\mathfrak {m}) \cdot \mathrm {ord}_{\mathfrak {m}}(f) = v(\mathfrak {m}^{\mathrm {ord}_{\mathfrak {m}}(f)}) \leq v(f)$. It remains to prove the existence of 
$c>0$ such that 
$v(f) \leq c \cdot \mathrm {ord}_{\mathfrak {m}}(f)$ for all 
$f\in R$.
If (i) holds, the existence of 
$c$ follows from Izumi's theorem for divisorial valuations as phrased in [Reference Rond and SpivakovskyRS14, Remark 1.6].Footnote 4 If (ii) holds, then there exists a log smooth birational model 
$\eta \in (Y,D)$ of 
$x\in X$ and 
$\boldsymbol {\alpha } \in \mathbb {R}^r$ such that 
$v=v_{\boldsymbol {\alpha }}$. Choose 
$\boldsymbol {\gamma } \in \mathbb {Z}_{>0}^r$ such that 
$\boldsymbol {\alpha }_i \leq \boldsymbol {\gamma }_i$ for each 
$i=1,\ldots, r$ and consider the valuation 
$w: = v_{\boldsymbol {\gamma }}$. We claim that 
$w\in \mathrm {DivVal}_{R,\mathfrak {m}}$. Assuming the claim, then (i) implies that there exists 
$c>0$ such that 
$w(f) \leq c \cdot \mathrm {ord}_{\mathfrak {m}}(f)$ for all 
$f\in R$. Since 
$v(f)\leq w(f)$ for all 
$f\in R$, (ii) then follows.
To verify that 
$w\in \mathrm {DivVal}_{R,\mathfrak {m}}$, note that 
$w: {\rm Frac\ }(R)^\times \to \mathbb {R}$ is the composition
\[ {\rm Frac}(R)^\times \hookrightarrow {\rm Frac}( \widehat{\mathcal{O}}_{Y,\eta})^\times \overset{\widehat{w}}{\longrightarrow } \mathbb{R}, \]
where 
$\widehat {w}$ is the valuation that sends 
$\sum _{\beta \in \mathbb {Z}^r_{\geq 0} } c_{\boldsymbol {\beta }} y^{\boldsymbol {\beta }}$ to 
$\min \{ \langle \boldsymbol {\alpha }, \boldsymbol {\beta } \rangle \mid c_{\boldsymbol {\beta }} \neq 0\}$. Using that 
$w$ is 
$\mathbb {Z}$-valued, a computation (see, for example, [Reference Jonsson and MustaţăJM12, Proposition 3.7]) shows 
$\widehat {w}$ is a divisorial valuation of 
$\widehat {\mathcal {O}}_{Y,\eta }$. Hence, 
$w$ is a divisorial valuation of 
$\mathcal {O}_{Y,\eta }$ by [Reference Huneke and SwansonHS06, Proposition 9.3.5]. Now, we compute
\[ {\rm tr.deg}_{\kappa}(\kappa_{w}) = {\rm tr.deg}_{\kappa}(k(\eta)) + {\rm tr.deg}_{k(\eta)}(\kappa_v) = {\rm tr.deg}_{\kappa}(k(\eta)) + (\dim \mathcal{O}_{Y,\eta} -1) = \dim R. \]
To see that the last equality holds, note that the complete local ring 
$\widehat {R}$ is a domain by assumption and thus equidimensional. Therefore 
$R$ is universally catenary by [Sta, Tag 0AW6] and so the dimension formula [Sta, Tag 02II] gives the last equality. Therefore, 
$w$ is divisorial as desired.
2.3 Intersection numbers
The theory of intersection numbers of line bundles on a proper scheme over an algebraically closed fields was developed in [Reference KleimanKle66]. We will use a more general framework developed in [Reference KleimanKle05, Appendix B].
2.3.1 Definition
Let 
$Z$ be a proper scheme over an Artinian ring 
$\Lambda$. For line bundles 
$\mathcal {L}_1,\ldots, \mathcal {L}_r$ on 
$Z$, the function
\[ \mathbb{Z}^r \ni (m_1,\ldots, m_r ) \to \sum_{i\geq0 } (-1)^i \ell_{\Lambda} \big( H^i \big(Z, \mathcal{L}_1^{m_1} \otimes \cdots \otimes \mathcal{L}_r^{m_r}\big) \big) \in \mathbb{Z} \]
is a polynomial of degree 
$\leq \dim Z$ [Reference KleimanKle05, Theorem B.7]. The intersection number 
$(\mathcal {L}_1 \cdot \cdots \mathcal {L}_r)$ is defined to be the coefficient of 
$m_1\cdots m_r$ in the above polynomial. When the choice of 
$\Lambda$ is unclear, we will write 
${(\mathcal {L}_1 \cdot \ldots \cdot \mathcal {L}_r)}_\Lambda$.
2.3.2 Intersections of exceptional divisors
Let 
$\mu :Y\to X=\mathrm {Spec}(R)$ be a proper birational morphism with 
$Y$ normal. For Cartier divisors 
$F_1,\ldots, F_{n-1}$ on 
$Y$ and a Weil divisor 
$D: = \sum _{i=1}^r a_i D_i$ on 
$Y$ with support contained in 
$Y_\kappa$, we set
\[ F_1 \cdot \ldots \cdot F_{n-1} \cdot D :=\sum_{i=1}^r a_i (\mathcal{O}_{Y}(F_1)\vert_{D_i} \cdot \ldots \cdot \mathcal{O}_{Y}(F_{n-1})\vert_{D_i})_\kappa . \]
This is well defined, since each prime divisor 
$D_i$ is proper over 
$\mathrm {Spec}(\kappa )$.
Proposition 2.5 Assume 
$R$ is complete. If 
$F_1, \ldots, F_n$ are Cartier divisors on 
$Y$ with support contained in 
$Y_\kappa$, then 
$F_1\cdot \ldots \cdot F_n$ is independent of the ordering of the 
$F_i$.
Since the intersection product in § 2.3.1 is symmetric, 
$F_1\cdot \ldots \cdot F_n$ is independent of the order of 
$F_1,\ldots, F_{n-1}$. To deduce the full result, we rely on intersection theory [Reference FultonFul98].
A subtle issue is that the results in [Reference FultonFul98, § 1-18] are stated for schemes of finite type over a field and, hence, do not immediately apply to 
$Y$. Fortunately, the Chow group of a scheme of finite type over a regular base scheme can be defined and the results of [Reference FultonFul98, § 2] extend to this setting by [Reference FultonFul98, §20.1] (see also [Sta, Chapter 02P3] for the results in an even more general setting).
Proof. By Cohen's structure theorem, there exists a surjective map 
$A \twoheadrightarrow R$, where 
$A$ is a regular local ring. Since the composition 
$Y\to X:= \mathrm {Spec}(R)\to \mathrm {Spec}(A)$ is finite type, the framework of [Reference FultonFul98, § 20.1] applies. Using intersection theory on 
$Y$ and its subschemes, we compute
\begin{align*} F_1 \cdot \ldots \cdot F_n &= \sum_{i=1}^r a_i \int_{D_i} c_1(\mathcal{M}_1) \cap \cdots \cap c_1(\mathcal{M}_{n-1}) \cap [D_i]\\ &= \int_{Y_\kappa } c_1(\mathcal{M}_1) \cap \cdots \cap c_1(\mathcal{M}_{n-1}) \cap [F_n] = \int_{Y_\kappa} F_1 \cdot \ldots \cdot F_{n-1} \cdot [F_n], \end{align*}
where 
$\int _{D_i}$ and 
$\int _{Y_\kappa }$ denote the degree maps induced by the proper morphisms 
$D_i \to \mathrm {Spec}(\kappa )$ and 
$Y_\kappa \to \mathrm {Spec}(\kappa )$ [Reference FultonFul98, Definition 1.4] and 
$\mathcal {M}_i : = \mathcal {O}_Y(F_i)$. The first equality holds by [Reference FultonFul98, Example 18.3.6] with the fact that 
$D_i$ is a proper scheme over 
$\kappa$, the second by the equality 
$[F_n] = \sum _{i=1}^r a_i [D_i]$, and the last by the definition of the first Chern class. Since
\[ F_1 \cdot \ldots \cdot F_{n-1} \cdot [F_n] = F_1 \cdot \ldots \cdot F_n \cdot [Y] \in { A}_*(Y_\kappa) \]
and the latter is independent of the order of the 
$F_i$ by [Reference FultonFul98, Corollary 2.4.2], the proposition holds.
2.4 Multiplicity
2.4.1 Multiplicity of an ideal
The multiplicity of an 
$\mathfrak {m}$-primary ideal 
$\mathfrak {a}$ is
\[ \mathrm{e}(\mathfrak{a}):= \lim_{m\to \infty} \frac{\ell(R/\mathfrak{a}^m)}{m^n/n!} . \]
The following intersection formula for multiplicities commonly appears in the literature when 
$x\in X$ is a closed point on a quasi-projective variety [Reference LazarsfeldLaz04, p. 92]. In the generality stated below, it follows from [Reference RamanujamRam73].
Proposition 2.6 Let 
$\mathfrak {a}\subset R$ be an 
$\mathfrak {m}$-primary ideal and 
$Y\to \mathrm {Spec}(R)$ a proper birational morphism with 
$Y$ normal. If 
$\mathfrak {a} \cdot \mathcal {O}_Y=\mathcal {O}_Y(-E)$ for a Cartier divisor 
$E = \sum _i a_i D_i$, then
\[ \mathrm{e}(\mathfrak{a})= (-E)^{n-1} \cdot E . \]
Proof. By [Reference RamanujamRam73, Theorem and Remark 1], 
$\mathrm {e}(\mathfrak {a}) = (\mathcal {L}^{n-1})_{R/\mathfrak {a}}$, where 
$\mathcal {L}:= \mathcal {O}_Y(-E)\vert _E$. Observe that
\[ (\mathcal{L}^{n-1})_{R/\mathfrak{a}} = \sum \ell_{\mathcal{O}_{E,\xi_i}}( \mathcal{O}_{E,\xi_i}) (\mathcal{L}_i^{n-1})_{R/\mathfrak{a}} =\sum a_i (\mathcal{L}_i^{n-1})_{R/\mathfrak{a}} , \]
where 
$E= \sum a_i D_i$ and 
$\mathcal {L}_{i} := \mathcal {L}\vert _{E_i}$. Indeed, the first equality is [Reference KleimanKle05, Lemma B.12] and the second equality holds, since 
$\mathcal {O}_{Y,\xi _i}$ is a DVR and 
$\mathcal {O}_{E,\xi _i} = \mathcal {O}_{Y,\xi }/ (\pi ^{a_i})$, where 
$\pi$ is a uniformizer of the DVR.
To finish the proof, notice that 
$D_i$ is defined over 
$\kappa$, since there is a natural inclusion 
$D_i \hookrightarrow Y_{\kappa,\mathrm {red}} \to Y_{\kappa }$. Since the residue field of the local ring 
$R/\mathfrak {a}$ is 
$\kappa$, any finite-length 
$\kappa$-module 
$M$ satisfies 
$\ell _{R/\mathfrak {a}}(M) = \ell _{\kappa }(M)$ by [Sta, Lemma 02M0]. Thus,
\[ \sum_{i=1}^r a_i (\mathcal{L}_i^{n-1})_{R/\mathfrak{a}} = \sum_{i=1}^r a_i (\mathcal{L}_i^{n-1})_{\kappa} = (-E)^{n-1} \cdot E , \]
which completes the proof.
2.4.2 Multiplicity of a filtration
Following [Reference Ein, Lazarsfeld and SmithELS03], the multiplicity of a graded sequence of 
$\mathfrak {m}$-primary ideals 
$\mathfrak {a}_\bullet$ is
\[ \mathrm{e}(\mathfrak{a}_\bullet):= \lim_{m\to\infty}\frac{\ell(R/\mathfrak{a}_m)}{m^n/n!} \in [0,\infty). \]
By [Reference Ein, Lazarsfeld and SmithELS03, Reference MustaMus02, Reference Lazarsfeld and MustaLM09, Reference CutkoskyCut13, Reference CutkoskyCut14] in increasing generality, the above limit exists and
\begin{equation} \mathrm{e}(\mathfrak{a}_\bullet)=\lim_{m\to\infty}\frac{\mathrm{e}(\mathfrak{a}_m)}{m^n} = \inf_{m} \frac{\mathrm{e}(\mathfrak{a}_m)}{m^n}; \end{equation}
see, in particular, [Reference CutkoskyCut14, Theorem 6.5]. Also defined in [Reference Ein, Lazarsfeld and SmithELS03], the volume of a valuation 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$ is
\[ \mathrm{vol}(v):= \mathrm{e}(\mathfrak{a}_\bullet(v)). \]
We set
\[ \mathrm{Val}_{R,\mathfrak{m}}^+:= \{v\in\mathrm{Val}_{R,\mathfrak{m}}\mid \mathrm{vol}(v)>0\}\subset \mathrm{Val}_{R,\mathfrak{m}} . \]
Proposition 2.7 Let 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$. If (i) 
$v$ is divisorial or (ii) 
$R$ contains a field and 
$v$ is quasi-monomial, then 
$\mathrm {vol}(v)>0$.
Proof. By Lemma 2.4, there exists 
$c>0$ such that 
$\mathfrak {a}_{\bullet }(v) \subset \mathfrak {m}^{c\bullet }$. Thus, 
$\mathrm {vol}(v)\geq \mathrm {e}(\mathfrak {m}^{c\bullet }) = c^{-n}\mathrm {e}(\mathfrak {m})>0$.
2.5 Normalized volume
Assume 
$R$ is the local ring of a closed point on a algebraic variety defined over an algebraically closed field of characteristic 0. We say 
$x\in (X,\Delta )$ is a klt singularity if 
$\Delta$ is an 
$\mathbb {R}$-divisor on 
$X$ such that 
$K_{X}+\Delta$ is 
$\mathbb {R}$-Cartier, and 
$(X,\Delta )$ is klt as defined in [Reference Kollár and MoriKM98].
The following invariant was first introduced in [Reference LiLi18] and plays an important role in the study of K-semistable Fano varieties and Fano cone singularities.
Definition 2.8 For a klt singularity 
$x\in (X,\Delta )$, the normalized volume function 
$\widehat {\mathrm {vol}}_{(X,\Delta ),x}:\mathrm {Val}_{X,x}\to (0,+\infty ]$ is defined by
\[ \widehat{\mathrm{vol}}_{(X,\Delta),x}(v):= \left\{\begin{array}{ll} A_{X,\Delta}(v)^n\cdot \mathrm{vol}(v), & \text{if } A_{X,\Delta}(v)<+\infty,\\ +\infty, & \text{if } A_{X,\Delta}(v)=+\infty, \end{array} \right. \]
where 
$A_{X,\Delta }(v)$ is the log discrepancy of 
$v$ as defined in [Reference Jonsson and MustaţăJM12, Reference Boucksom, de Fernex, Favre and UrbinatiBdFFU15]. The local volume of a klt singularity 
$x\in (X,\Delta )$ is defined as
\[ \widehat{\mathrm{vol}}(x,X,\Delta):= \inf_{v\in\mathrm{Val}_{X,x}} \widehat{\mathrm{vol}}_{(X,\Delta),x}(v). \]
The above infimum is indeed a minimum by [Reference BlumBlu18, Reference XuXu20].
3. Saturation
Throughout this section, 
$(R,\mathfrak {m},\kappa )$ denotes an 
$n$-dimensional, analytically irreducible, Noetherian, local domain.
3.1 Definition of the saturation
Let 
$\mathfrak {a}\subset R$ be an 
$\mathfrak {m}$-primary ideal. Recall that the integral closure 
$\overline {\mathfrak {a}}$ of 
$\mathfrak {a}$ can be characterized valuatively by
\[ \overline{\mathfrak{a}}=\{f\in R\mid v(f)\ge v(\mathfrak{a}) \text{ for all } v\in\mathrm{DivVal}_{R,\mathfrak{m}}\}. \]
See, for example, [Reference Huneke and SwansonHS06, Theorem 6.8.3] and [Reference LazarsfeldLaz04, Example 9.6.8]. We define the saturation of an 
$\mathfrak {m}$-filtration in a similar manner.
Definition 3.1 The saturation 
$\widetilde {\mathfrak {a}}_\bullet$ of an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ is defined by
\[ \widetilde{\mathfrak{a}}_\lambda:= \{f\in \mathfrak{m} \mid v(f)\ge \lambda\cdot v(\mathfrak{a}_\bullet) \text{ for all } v\in\mathrm{DivVal}_{R,\mathfrak{m}} \} \]
for each 
$\lambda \in \mathbb {R}_{>0}$. We say 
$\mathfrak {a}_\bullet$ is saturated if 
$\mathfrak {a}_\bullet =\widetilde {\mathfrak {a}}_\bullet$.
Remark 3.2 Proposition 3.19 shows that it is equivalent to define the saturation using all positive volume valuations, rather than only divisorial valuations.
Remark 3.3 The saturation is a local analogue of the maximal norm of a multiplicative norm of the section ring of a polarized variety, which was defined and studied in [Reference Boucksom and JonssonBJ21]. See § 6.
Remark 3.4 Definition 3.1 differs from the definition of saturation used in [Reference MustaMus02, § 2] for monomial ideals, which coincides with the ideals in Lemma 3.6 defined using the integral closure of the Rees algebra.
Remark 3.5 After the first version of this paper was posted on the arXiv, Cutkosky and Praharaj introduced an operation on 
$\mathbb {R}$-filtrations that is defined using certain asymptotic Hilbert–Samuel functions (see the definition in [Reference Cutkosky and PraharajCP22, Theorem 1.3]). As shown by [Reference Cutkosky and PraharajCP22, Example 7.2], their operation does not always coincide with the saturation.
Lemma 3.6 [Reference CutkoskyCut21, Lemma 5.6]
If 
$\mathfrak {a}_\bullet$ be an 
$\mathfrak {m}$-filtration, then the integral closure of 
${\rm Rees}(\mathfrak {a}_\bullet ) := \bigoplus _{m \in \mathbb {N}} \mathfrak {a}_m t^m$ in 
$R[t]$ is given by 
$\bigoplus _{m\in \mathbb {N}} {\mathfrak {a}}'_m t^m$, where
\[ {\mathfrak{a}}'_m=\{f\in R \mid\text{there exists } r>0 \text{ such that } f^r\in \overline{\mathfrak{a}}_{rm}\}. \]
Example 3.7 In general, the ideals 
$\widetilde {\mathfrak {a}}_m$ and 
${\mathfrak {a}}'_m$ appearing above do not coincide. For example, let 
$R:= k[x]_{(x)}$ and 
$\mathfrak {m}=(x)$. Consider the 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ defined by 
$\mathfrak {a}_\lambda := (x^{\lceil \lambda +1\rceil })$. Using that 
$\mathrm {ord}_{\mathfrak {m}} (\mathfrak {a}_\lambda ) =\lceil \lambda +1 \rceil$ and 
$\mathrm {ord}_{\mathfrak {m}} (\mathfrak {a}_\bullet )= 1$, we compute
\[ {\mathfrak{a}}'_m = (x^{m+1}) \subsetneq (x^m) = \widetilde{\mathfrak{a}}_m \]
for each 
$m\in \mathbb {Z}_{>0}$. In particular, 
${\rm Rees}(\mathfrak {a}_\bullet )$ is integrally closed, but 
$\mathfrak {a}_\bullet$ is not saturated.
The following lemma states basic properties of the saturation.
Lemma 3.8 For any 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$, the following statements hold:
(1)
$\mathfrak {a}_\bullet \subset \widetilde {\mathfrak {a}}_\bullet$;(2)
$v(\mathfrak {a}_\bullet )=v(\widetilde {\mathfrak {a}}_\bullet )$ for all $v\in \mathrm {DivVal}_{R,\mathfrak {m}}$; and(3)
$\widetilde {\mathfrak {a}}_\bullet$ is saturated.
Proof. For any 
$\lambda \in \mathbb {R}_{>0}$,
\[ v(\mathfrak{a}_\lambda) = \lim_{m \to \infty} \frac{ v(\mathfrak{a}_\lambda^m)}{m} \geq \lim_{m\to\infty} \biggl( \frac{ v\big(\mathfrak{a}_{\lfloor \lambda m\rfloor }\big)}{\lfloor \lambda m\rfloor } \cdot \frac{ \lfloor \lambda m \rfloor}{m} \biggr)= \lambda v(\mathfrak{a}_\bullet) , \]
where the inequality uses that 
$(\mathfrak {a}_{\lambda })^m \subset \mathfrak {a}_{\lambda m} \subset \mathfrak {a}_{ \lfloor \lambda m \rfloor }$. Therefore, 
$\mathfrak {a}_{\lambda } \subset \widetilde {\mathfrak {a}}_\lambda$, which is (1).
For (2), note that 
$v(\mathfrak {a}_\bullet ) \leq {v(\widetilde {\mathfrak {a}}_m ) }/{m} \leq {v(\mathfrak {a}_m)}/{m}$ for each 
$m \in \mathbb {Z}_{>0}$, where the first inequality follows from the definition of 
$\widetilde {\mathfrak {a}}_m$ and the second from (1). Sending 
$m\to \infty$ gives 
$v(\mathfrak {a}_\bullet ) \leq v(\widetilde {\mathfrak {a}}_\bullet )\leq v(\mathfrak {a}_\bullet )$, which implies (2). Statement (3) follows immediately from (2).
We say two 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ are equivalent if 
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_\bullet$. The following proposition gives a characterization of when two filtrations are equivalent after possible scaling.
Proposition 3.9 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations and 
$c\in \mathbb {R}_{>0}$. The following statements are equivalent.
(1)
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_{c \bullet }$.(2)
$v(\mathfrak {a}_\bullet ) = c v(\mathfrak {b}_\bullet )$ for all $v\in \mathrm {DivVal}_{R,\mathfrak {m}}$.
Proof. First, note that 
$v(\mathfrak {b}_{c \bullet }) =c v(\mathfrak {b}_\bullet )$ for all 
$v\in \mathrm {Val}_{R,\mathfrak {m}}$. Therefore, (1) implies (2) follows from Lemma 3.8(2), while (2) implies (1) follows from the definition of the saturation.
3.2 Saturation and completion
Let 
$(\widehat {R},\widehat {\mathfrak {m}})$ denote the 
$\mathfrak {m}$-adic completion of 
$(R,\mathfrak {m})$ and write 
$\varphi :R\to \hat {R}$ for the natural morphism. (Note that 
$\widehat {R}$ is a domain by the assumption throughout the paper that 
$R$ is analytically irreducible.) For an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$, we set 
$\mathfrak {a}_\bullet \widehat {R} : = (\mathfrak {a}_\lambda \widehat {R})_{\lambda >0}$, which is an 
$\widehat {\mathfrak {m}}$-filtration.
Proposition 3.10 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration and 
$\mathfrak {b}_\bullet :=\mathfrak {a}_\bullet \widehat {R}$, then 
$\widetilde {\mathfrak {a}}_\bullet \widehat {R} = \widetilde {\mathfrak {b}}_\bullet$.
The proposition shows that completion and saturation commute, as is the case with the integral closure of ideals [Reference Huneke and SwansonHS06, Proposition 1.6.2]. As a consequence of the proposition, many results regarding saturations reduce to the case when 
$R$ is a complete local domain.
Proof. By [Reference Huneke and SwansonHS06, Theorem 9.3.5], there is a bijective map 
$\varphi _* \colon \mathrm {DivVal}_{\widehat {R},\widehat {\mathfrak {m}}} \to \mathrm {DivVal}_{R,\mathfrak {m}}$ that sends 
$\hat {v}$ to the valuation 
$v$ defined by composition
\[ {\rm Frac}(R)^\times \hookrightarrow {\rm Frac}(\hat{R})^\times \overset{\hat{v}}{\to} \mathbb{R} . \]
Note that 
$\hat {v}(\mathfrak {a} \hat {R}) = v(\mathfrak {a})$ for any ideal 
$\mathfrak {a}\subset R$.
Fix 
$\lambda \in \mathbb {R}_{>0}$. Using the previous observation twice and the definition of the saturation, we compute
\[ \hat{v} (\widetilde{\mathfrak{a}}_\lambda \hat{R} ) = v(\widetilde{\mathfrak{a}}_\lambda) \geq \lambda v (\mathfrak{a}_\bullet) = \lambda \hat{v}(\mathfrak{b}_\bullet) . \]
Thus, 
$\widetilde {\mathfrak {a}}_{\lambda } \hat {R} \subset \widetilde {\mathfrak {b}}_\lambda$. To prove the reverse inclusion, note that 
$\mathfrak {c} R = \widetilde {\mathfrak {b}}_\lambda$ for some ideal 
$\mathfrak {c}\subset R$, since 
$\widetilde {\mathfrak {b}}_\lambda$ is 
$\widehat {\mathfrak {m}}$-primary. As before, we compute
\[ v(\mathfrak{c})= \hat{v}(\widetilde{\mathfrak{b}}_\lambda) \geq \lambda \hat{v}(\mathfrak{b}_\bullet) =\lambda v (\mathfrak{a}_\bullet) . \]
Therefore, 
$\widetilde {\mathfrak {b}}_\lambda = \mathfrak {c} \hat {R} \subset \widetilde {\mathfrak {a}}_\lambda \hat {R}$.
Proposition 3.11 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration and 
$\mathfrak {b}_\bullet = \mathfrak {a}_\bullet \widehat {R}$, then 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\mathfrak {b}_\bullet )$.
Proof. Since 
$R/\mathfrak {a}_\lambda$ and 
$\widehat {R}/\mathfrak {b}_\lambda$ are isomorphic as Artinian rings, 
$\ell (R/\mathfrak {a}_\lambda )=\ell (\widehat {R}/\mathfrak {b}_\lambda )$ for each 
$\lambda >0$. Therefore, the equality of multiplicities holds.
3.3 Saturation and multiplicity
In this section we prove Theorem 1.4 and a number of corollaries. The theorem is a consequence of the following two propositions.
Proposition 3.12 Let 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. If 
$\mathrm {e}(\mathfrak {a}_\bullet )= \mathrm {e}(\mathfrak {b}_\bullet )$, then 
$v(\mathfrak {a}_\bullet ) = v(\mathfrak {b}_\bullet )$ for all 
$v\in \mathrm {Val}_{R,\mathfrak {m}}^+$.
The proposition was shown when 
$v$ is divisorial in [Reference CutkoskyCut21, Theorem 7.3] using Okounkov bodies. The proof below instead follows the strategy of [Reference Li and XuLX20, Proposition 2.7] and [Reference Xu and ZhuangXZ21, Lemma 3.9].
Proof. Suppose the statement is false. Then there exists 
$v\in \mathrm {Val}_{R,\mathfrak {m}}^+$ such that 
$v(\mathfrak {b}_\bullet ) < v(\mathfrak {a}_\bullet )$. After scaling 
$v$, we may assume 
$v(\mathfrak {a}_\bullet )=1$. Therefore, there exists 
$l \in \mathbb {Z}_{>0}$ so that 
$v(\mathfrak {b}_{l}) /l < v(\mathfrak {a}_\bullet )=1$. Hence, we may choose 
$f\in \mathfrak {b}_l\setminus \mathfrak {a}_l$ such that 
$k:= v(f)< l$.
Now, consider the map
\[ \phi: R \to \mathfrak{b}_{l m}/ \mathfrak{a}_{lm}\quad \text{sending}\quad g \mapsto f^m g. \]
We claim 
$\ker (\phi ) \subset \mathfrak {a}_{ (l-k)m } (v)$. Indeed, if 
$g\in \ker \phi$, then 
$f^m g \in \mathfrak {a}_{lm}$. Hence,
\[ v(g) =v(f^m g) - v(f^m) \geq v(\mathfrak{a}_{l m}) - km \geq l m - km = (l-k) m \]
as desired. Using the claim, we deduce
\begin{equation} \ell( \mathfrak{b}_{l m}/ \mathfrak{a}_{lm}) \geq \ell (R/ \ker \phi) \geq \ell ( R/ \mathfrak{a}_{ (l-k)m } (v)) . \end{equation}Finally, we compute
\[ \mathrm{e}(\mathfrak{a}_\bullet) - \mathrm{e}(\mathfrak{b}_\bullet) = \lim_{ m \to \infty} \frac{ \ell (\mathfrak{b}_{lm } / \mathfrak{a}_{lm})}{(lm)^n /n!} \geq \lim_{m\to\infty} \frac{\ell ( R/ \mathfrak{a}_{ (l-k) m }(v) )}{ (lm)^n /n!} = \frac{(l-k)^n \,\mathrm{vol}(v)}{l^n} >0 , \]
where the first inequality is by (3.1) and the second uses that 
$v$ has positive volume. This contradicts our assumption that 
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\mathfrak {b}_\bullet )$.
Proposition 3.13 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. Assume 
$(R,\mathfrak {m})$ is complete. If 
$v(\mathfrak {b}_\bullet ) \leq v(\mathfrak {a}_\bullet )$ for all 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$, then 
$\mathrm {e}(\mathfrak {b}_\bullet ) \leq \mathrm {e}(\mathfrak {a}_\bullet )$.
The proposition and its proof are local analogues of [Reference SzékelyhidiSzé15, Lemma 22], which concerns the volumes of graded linear series of projective varieties.
Remark 3.14 The assumption that 
$R$ is complete in Proposition 3.13 implies that 
$R$ is Nagata [Sta, Lemma 032W] (see also [Sta, Definition 033S]). The latter property will be used in the proof of Proposition 3.13 to ensure certain normalization morphisms are proper.
Proof. We claim that there exists a sequence of proper birational morphisms
\[ \cdots\to X_3 \to X_2\to X_1\to X_0 = X \]
such that the each 
$X_i$ is normal and the sheafs 
$\mathfrak {a}_{l} \cdot \mathcal {O}_{X_m}$ and 
$\mathfrak {b}_{l}\cdot \mathcal {O}_{X_m}$ are line bundles when 
$l\leq m$. Such a sequence can be constructed inductively as follows. Let 
$X_{i} \to X_{i-1}$ be defined by the composition
\[ X_{i} \to X_{i,2}\to X_{i,1} \to X_{i-1}, \]
where 
$X_{i,1}$ is the blowup of 
$\mathfrak {a}_i \cdot \mathcal {O}_{X_{i-1}}$, 
$X_{i,2}$ is the blowup of 
$\mathfrak {b}_i \cdot \mathcal {O}_{X_{i,1}}$, and 
$X_i$ the normalization of 
$X_{i,2}$. Note that 
$X_{i,2} \to X$ is proper, since 
$X_{i,2}\to X_{i-1}$ is a blowup and 
$X_{i-1}\to X$ is proper by our inductive assumption. Since 
$X$ is Nagata and 
$X_{i,2}\to X$ is finite type, 
$X_{i,2}$ is Nagata by [Sta, Lemma 035A]. Therefore, 
$X_i\to X_{i,2}$ is finite by the definition of Nagata, and we conclude the composition 
$X_i \to X_{i-1}$ is proper, which completes the proof of the claim.
Next, consider a proper birational morphism 
$Y\to X$ with 
$Y$ normal and factoring as
\[ Y \overset{\pi_m}{\longrightarrow} X_m \longrightarrow X, \]
For 
$l\leq m$, there exist Cartier divisors 
$G_l$ and 
$G'_l$ on 
$Y$ such that
\[ \mathfrak{a}_l \cdot \mathcal{O}_{Y} = \mathcal{O}_{Y} (G_l) \quad \text{ and } \quad \mathfrak{b}_l\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(G_l') . \]
Set 
$F_l:= {G_l}/{l}$ and 
$F'_l:= {G'_l}/{l}$, which are relatively nef over 
$X$ and satisfy
\[ F_l=- \sum_{E \subset Y} \frac{\mathrm{ord}_E(\mathfrak{a}_l)}{l} E \quad \text{ and }\quad F'_l =- \sum_{E \subset Y} \frac{\mathrm{ord}_E(\mathfrak{b}_l)}{l} E, \]
where the sums run through prime divisors 
$E\subset Y$. Throughout the proof, we will without mention replace 
$Y$ with higher birational models so it factors through certain 
$X_m\to X$.
Given 
$\epsilon >0$, by Proposition 2.6 and (2.1), there exists 
$m_0>0$ such that
\begin{equation} - F_{m_0}^{n-1} \cdot F_{m_0} =\mathrm{e}(\mathfrak{a}_{m_0})/{m_0}^n< \mathrm{e}(\mathfrak{a}_\bullet) + \epsilon. \end{equation}
For a multiple 
$m_1$ of 
$m_0$,
\[ -(F_{m_0})^{n-1} \cdot F_{m_1}< \mathrm{e}(\mathfrak{a}_\bullet) + \epsilon , \]
since 
$F_{m_0} \leq F_{m_1}$ and 
$F_{m_0}$ is nef over 
$X$. Now, we compute
\begin{align*} - (F_{m_0})^{n-1} \cdot
F_{m_1} + (F_{m_0})^{n-1} \cdot F'_{m_1} &= \sum_{E\subset
Y } \big(\!- \mathrm{ord}_{E} (F_{m_1}) +\mathrm{ord}_{E}
(F'_{m_1}) \big) (F_{m_0})^{n-1} \cdot E\\ &=
\sum_{E\subset X_{m_0} } \big(\!-\mathrm{ord}_{E} (F_{m_1})
+\mathrm{ord}_{E} (F'_{m_1}) \big) ({\pi_{m_0}}_*
F_{m_0})^{n-1} \cdot E, \end{align*}
where the second equality uses that 
$F_{m_0}$ is the pullback of 
${\pi _{m_0}}_* F_{m_0}$ and the projection formula [Reference KleimanKle05, Proposition B.16]. Since 
${\pi _{m_0}}_* F_{m_0}$ is nef over 
$X$ and
\[ \lim_{m\to \infty} \big(\!-\mathrm{ord}_{E} (F_m) + \mathrm{ord}_{E}(F'_m) \big) = \mathrm{ord}_E(\mathfrak{a}_\bullet) -\mathrm{ord}_E(\mathfrak{b}_\bullet) \geq 0 , \]
we may choose 
$m_1$ sufficiently large and divisible by 
$m_0$ such that
\[ -(F_{m_0})^{n-1} \cdot F'_{m_1} \leq - F_{m_0}^{n-1} \cdot F_{m_1} + \epsilon. \]
Therefore,
\[ - (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_0} = - (F_{m_0})^{n-1} \cdot F'_{m_1} \leq \mathrm{e}(\mathfrak{a}_\bullet) + 2\epsilon , \]
where the first equality is Proposition 2.5.
For any multiple 
$m_2$ of 
$m_1$,
\[ - (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_2} \leq \mathrm{e}(\mathfrak{a}_\bullet) + 2\epsilon, \]
since 
$F_{m_0} \leq F_{m_2}$ and the terms 
$F'_{m_1}$ and 
$F_{m_0}$ are nef over 
$X$. Similarly to the previous paragraph, we compute
\begin{align*} &- (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_2} - (F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F'_{m_2}\\ &\qquad =\sum_{E \subset X_{m_1} } \big(\!- \mathrm{ord}_{E} (F_{m_2})+ \mathrm{ord}_{E} (F'_{m_2}) \big) ({ ({\pi_{m_1}}_* F_{m_0})^{n-1} \cdot \pi_{m_1}}_*F'_{m_1}) \cdot E. \end{align*}
and, hence, we may choose 
$m_2$ sufficiently large and divisible by 
$m_1$ so that
\[ -(F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F'_{m_2} <-(F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F_{m_2}+ \epsilon. \]
Thus,
\[ -(F_{m_0})^{n-2} \cdot F'_{m_1} \cdot F'_{m_2} < \mathrm{e}(\mathfrak{a}_\bullet) +3\epsilon. \]
Repeating in this way gives
\[ -F'_{m_1} \cdot F'_{m_2} \cdot \cdots \cdot F'_{m_n} < \mathrm{e}(\mathfrak{a}_\bullet)+ (n+1)\epsilon , \]
where each 
$m_i$ divides 
$m_{i+1}$. Since each 
$F'_{m_i}$ is nef over 
$X$ and 
$F'_{m_{i}} \leq F'_{m_{i+1}}$, we see that
\[ -(F'_{m_n})^n < \mathrm{e}(\mathfrak{a}_\bullet)+ n\epsilon. \]
Using that 
$\mathrm {e}(\mathfrak {b}_\bullet ) = \inf _{m} \mathrm {e}(\mathfrak {b}_m)/m^n =\inf _{m} -(F'_{m})^n$ by Lemma 2.6 and (2.1), we see that 
$\mathrm {e}(\mathfrak {b}_\bullet ) < \mathrm {e}(\mathfrak {a}_\bullet ) + (n+1)\epsilon$. Therefore, 
$\mathrm {e}(\mathfrak {b}_\bullet )\leq \mathrm {e}(\mathfrak {a}_\bullet )$.
Remark 3.15 When 
$x\in X$ is an isolated singularity on a normal variety, Proposition 3.13 follows easily from the intersection theory for nef 
$b$-divisors developed in [Reference Boucksom, de Fernex and FavreBdFF12]. Indeed, the assumption 
$v(\mathfrak {a}_\bullet ) \leq v(\mathfrak {b}_\bullet )$ implies 
$Z(\mathfrak {a}_\bullet )\ge Z(\mathfrak {b}_\bullet )$, where 
$Z(\mathfrak {a}_\bullet )$ is the nef 
$b$-Weil 
$\mathbb {R}$-divisor associated to 
$\mathfrak {a}_\bullet$. The proposition then follows from [Reference Boucksom, de Fernex and FavreBdFF12, Remark 4.17]. However, when 
$x$ is not an isolated singularity, a satisfactory intersection theory for nef 
$b$-divisors seems missing from the literature.
We are now ready to prove Theorem 1.4 and a number of its corollaries.
Proof of Theorem 1.4 By Propositions 3.10 and 3.11, we may assume 
$(R,\mathfrak {m})$ is complete. This condition will be needed below to apply Proposition 3.13.
By Propositions 3.12 and 3.13, 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\mathfrak {b}_\bullet )$ if and only if 
$v(\mathfrak {a}_\bullet ) = v(\mathfrak {b}_\bullet )$ for all 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$. By Proposition 3.9, the latter condition holds if and only if 
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_\bullet$.
Corollary 3.16 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration, then 
$\mathrm {e}(\mathfrak {a}_\bullet )=\mathrm {e}(\widetilde {\mathfrak {a}}_\bullet )$.
Proof. By Lemma 3.8, 
$\mathfrak {a}_\bullet \subset \widetilde {\mathfrak {a}}_\bullet$ and 
$\widetilde {\mathfrak {a}}$ is saturated. Thus, Theorem 1.4 implies 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\widetilde {\mathfrak {a}}_\bullet )$.
Corollary 3.17 Let 
$\mathfrak {a}_\bullet$ and 
$\mathfrak {b}_\bullet$ be 
$\mathfrak {m}$-filtrations. The following statements are equivalent.
(1)
$\widetilde {\mathfrak {a}}_\bullet = \widetilde {\mathfrak {b}}_{ \bullet }$.(2)
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\mathfrak {a}_\bullet \cap \mathfrak {b}_{\bullet })= \mathrm {e}(\mathfrak {b}_\bullet )$.
Proof of Corollary 3.17 Assume (1) holds. Observe that
\[ \mathrm{e}(\mathfrak{a}_\bullet)=\mathrm{e}(\widetilde{\mathfrak{a}}_\bullet) = \mathrm{e}(\widetilde{\mathfrak{b}}_{\bullet}) = \mathrm{e}(\mathfrak{b}_\bullet) , \]
where the first and third equality follow from Corollary 3.16. Thus, it remains to show that 
$\mathrm {e}(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet )= \mathrm {e}(\mathfrak {b}_\bullet )$. First, note that 
$\mathrm {e}(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet ) \geq \mathrm {e}(\mathfrak {b}_\bullet )$ holds trivially, since 
$\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet \subset \mathfrak {b}_\bullet$. For the reverse inequality, we compute
\begin{align*} \mathrm{e}(\mathfrak{a}_\bullet\cap \mathfrak{b}_{\bullet}) = \lim_{m \to \infty} \frac{ \ell (R/(\mathfrak{a}_m \cap \mathfrak{b}_{ m}))}{m^n/n!} &= \lim_{m \to \infty} \biggl( \frac{ \ell (R/(\mathfrak{a}_m) )}{m^n/n!} + \frac{\ell(R/\mathfrak{b}_{ m}) }{m^n/n!}-\frac{ \ell(R/(\mathfrak{a}_m+\mathfrak{b}_{ m}))}{m^n/n!}\biggr)\\ &= \mathrm{e}(\mathfrak{a}_\bullet) +\mathrm{e}(\mathfrak{b}_{ \bullet}) - \lim_{m\to \infty} \frac{ \ell(R/(\mathfrak{a}_m+\mathfrak{b}_{ m}))}{m^n/n!}\\ &\leq \mathrm{e}(\mathfrak{a}_\bullet) +\mathrm{e}(\mathfrak{b}_{ \bullet}) - \lim_{m\to \infty} \frac{ \ell(R/\widetilde{\mathfrak{a}}_m)}{m^n/n!}\\ &= \mathrm{e}(\mathfrak{a}_\bullet) +\mathrm{e}(\mathfrak{b}_{ \bullet}) - \mathrm{e}(\widetilde{\mathfrak{a}}_\bullet)\\ & = \mathrm{e}(\mathfrak{b}_\bullet), \end{align*}
where the inequality uses that 
$\mathfrak {a}_m+\mathfrak {b}_m \subset \widetilde {\mathfrak {a}}_m + \widetilde {\mathfrak {b}}_m = \widetilde {\mathfrak {a}}_m+\widetilde {\mathfrak {a}}_m = \widetilde {\mathfrak {a}}_m$ by the assumption that (1) holds. Therefore, (2) holds.
Conversely, assume (2) holds. Applying Theorem 1.4 to both 
$(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet ) \subset \mathfrak {a}_\bullet$ and 
$(\mathfrak {a}_\bullet \cap \mathfrak {b}_\bullet )\subset \mathfrak {b}_\bullet$, we see that 
$\widetilde {\mathfrak {a}_\bullet \cap \mathfrak {b}_{ \bullet }}= \widetilde {\mathfrak {a}}_\bullet$ and 
$\widetilde {\mathfrak {a}_\bullet \cap \mathfrak {b}_{ \bullet }}= \widetilde {\mathfrak {b}}_\bullet$, which implies that (1) holds.
The following result was proven when 
$R$ is regular in [Reference MustaMus02, Theorem 1.7.2].
Corollary 3.18 An 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ is linearly bounded if and only if 
$\mathrm {e}(\mathfrak {a}_\bullet )>0$.
Proof. If 
$\mathfrak {a}_\bullet$ is linearly bounded, then there exists 
$c>0$ such that 
$\mathfrak {a}_{\lambda } \subset \mathfrak {m}^{\lceil c \lambda \rceil }$ for all 
$\lambda >0$. Thus, 
$\mathrm {e}(\mathfrak {a}_\bullet ) \geq c^{n} \mathrm {e}(\mathfrak {m}) >0$ as desired.
Next, assume 
$\mathrm {e}(\mathfrak {a}_\bullet )>0$. We claim that there exists 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$ such that 
$v(\mathfrak {a}_\bullet )>0$. If not, then 
$\widetilde {\mathfrak {a}}_\lambda = \mathfrak {m}$ for all 
$\lambda >0$. Using Corollary 3.16, we then see 
$\mathrm {e}(\mathfrak {a}_\bullet ) = \mathrm {e}(\widetilde {\mathfrak {a}}_\bullet ) =0$, which is a contradiction. Now, fix 
$v\in \mathrm {DivVal}_{R,\mathfrak {m}}$ with 
$v(\mathfrak {a}_\bullet ) >0$. Using that 
$\mathfrak {a}_\bullet \subset \mathfrak {a}_{\bullet }(v)$ and Proposition 2.7, we conclude 
$\mathrm {e}(\mathfrak {a}_\bullet )\geq \mathrm {e}(\mathfrak {a}_\bullet (v)) >0$.
3.4 Saturation and finite-volume valuations
Using results from the previous section, we show that the saturation can be defined using positive volume valuations, rather than only divisorial valuations.
Proposition 3.19 If 
$\mathfrak {a}_\bullet$ is an 
$\mathfrak {m}$-filtration and 
$\lambda \in \mathbb {R}_{>0}$, then
\[ \widetilde{\mathfrak{a}}_\lambda = \big\{ f\in \mathfrak{m} \mid v(f) \geq \lambda \cdot v(\mathfrak{a}_\bullet) \text{ for all } v\in \mathrm{Val}_{R,\mathfrak{m}}^+ \big\}. \]
The proposition will be deduced from the following lemma.
Lemma 3.20 If 
$\{ v_i \}_{i\in I}$ is a collection of valuations in 
$\mathrm {Val}_{R,\mathfrak {m}}^+$ and 
$\{ c_i \}_{i\in I}$ of non-negative real numbers, then the 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_\bullet$ defined by
\[ \mathfrak{a}_\lambda := \{ f\in \mathfrak{m} \mid v_i (f) \geq \lambda c_i \text{ for all } i \in I\} \]
is saturated. Hence, 
$\mathfrak {a}_\bullet (v)$ is saturated for any 
$v\in \mathrm {Val}_{R,\mathfrak {m}}^+$.
Proof. Suppose the statement is false. Then there exists some 
$\lambda \in \mathbb {R}_{>0}$ such that 
$\mathfrak {a}_\lambda \subsetneq \widetilde {\mathfrak {a}}_{\lambda }$. Thus, there exist 
$f\in \widetilde {\mathfrak {a}}_{\lambda }$ and 
$i \in I$ such that 
$v_i(f) < \lambda c_i$. We claim 
$v_i(\widetilde {\mathfrak {a}}_\bullet )< v_i(\mathfrak {a}_\bullet )$. Indeed,
\[ v_i(\widetilde{\mathfrak{a}}_\bullet)= \lim_{m \to \infty} \frac{ v_i(\widetilde{\mathfrak{a}}_{ \lfloor \lambda m\rfloor })}{ \lfloor \lambda m \rfloor } \leq \lim_{m\to \infty} \frac{ v_i(f^m)}{ \lfloor \lambda m \rfloor } = \frac{ v_i(f) }{\lambda}< c_i, \]
where the second equality uses that 
$f^m \in (\widetilde {\mathfrak {a}}_{\lambda })^m \subset \widetilde {\mathfrak {a}}_{\lambda m} \subset \widetilde {\mathfrak {a}}_{\lfloor \lambda m \rfloor }$. Since 
$v_i(\mathfrak {a}_\bullet ) \geq c_i$, the claim holds.
By Proposition 3.12 and the claim, 
$\mathrm {e}(\widetilde {\mathfrak {a}}_\bullet )< \mathrm {e}(\mathfrak {a}_\bullet )$. The latter contradicts Theorem 1.4.
Proof of Proposition 3.19 Let 
$\mathfrak {b}_\bullet$ denote the 
$\mathfrak {m}$-filtration defined by
\[ \mathfrak{b}_\lambda := \{f \in \mathbb{R} \mid v(f) \geq \lambda \cdot v(\mathfrak{a}_\bullet) \text{ for all } v \in \mathrm{Val}_{R,\mathfrak{m}}^+ \} . \]
Notice that 
$\mathfrak {a}_\bullet \subset \mathfrak {b}_\bullet \subset \widetilde {\mathfrak {a}}_\bullet$, where the second inclusion uses that all divisorial valuations have positive volume. Taking saturations gives 
$\widetilde {\mathfrak {a}}_\bullet \subset \widetilde {\mathfrak {b}}_\bullet \subset \widetilde {\widetilde {\mathfrak {a}}}_\bullet$. Since 
$\mathfrak {b}_\bullet$ and 
$\widetilde {\mathfrak {a}}_\bullet$ are saturated by Lemma 3.20, we conclude 
$\widetilde {\mathfrak {a}}_\bullet \subset \mathfrak {b}_\bullet \subset {\widetilde {\mathfrak {a}}}_\bullet$.
4. Multiplicity and geodesics
In this section, we prove Theorem 1.1 on the convexity of the multiplicity function along geodesics. Throughout, 
$(R,\mathfrak {m},\kappa )$ denotes an 
$n$-dimensional, analytically irreducible, Noetherian local domain containing a field of arbitrary characteristic.
4.1 Geodesics
Fix two finite-multiplicity 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$.
Definition 4.1 For each 
$t\in (0,1)$, we define an 
$\mathfrak {m}$-filtration 
$\mathfrak {a}_{\bullet,t}$ by setting
\[ \mathfrak{a}_{\lambda,t} = \sum_{\lambda=(1-t)\mu+t\nu} \mathfrak{a}_{\mu,0} \cap \mathfrak{a}_{\nu,1} , \]
where the sum runs through all 
$\mu, \nu \in \mathbb {R}$ satisfying 
$\lambda =(1-t)\mu +t\nu$. We call 
$(\mathfrak {a}_{\bullet,t})_{t\in [0,1]}$ the geodesic between 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$.
This definition is a local analog of the geodesic between two filtrations of the section ring of a polarized variety [Reference Blum, Liu, Xu and ZhuangBLXZ23, Reference RebouletReb22]. See Section 6 for details.
Example 4.2 Let 
$R:= k[x,y]_{(x,y)}$. For 
$\boldsymbol {\alpha }=(\boldsymbol {\alpha }_1,\boldsymbol {\alpha }_2) \in \mathbb {R}_{>0}^2$, let 
$v_{\boldsymbol {\alpha }}\colon {\rm Frac}(R)^\times \to \mathbb {R}$ be the monomial valuation with weight 
$\boldsymbol {\alpha }_1$ and 
$\boldsymbol {\alpha }_2$ with respect to 
$x$ and 
$y$, that is,
\[ v\biggl( \sum_{m,n \geq 0 } c_{m,n} x^{m}y^{n} \biggr) = \min \{ \boldsymbol {\alpha}_1m + \boldsymbol {\alpha}_2 n \mid c_{m,n} \neq 0\}. \]
Note that 
$\mathfrak {a}_\lambda (v_{\boldsymbol {\alpha }}) = \{x^{m}y^n \mid m \alpha _1 + n \alpha _2 \geq \lambda \}$. Now, fix 
$\boldsymbol {\alpha }, \boldsymbol {\beta } \in \mathbb {R}_{\geq 0}^2$ and consider the geodesic 
$\mathfrak {a}_{\bullet,t}$ between 
$\mathfrak {a}_{\bullet,0} := \mathfrak {a}_{\bullet }(v_{\boldsymbol {\alpha }})$ and 
$\mathfrak {a}_{\bullet,1}:= \mathfrak {a}_{\bullet }(v_{\boldsymbol {\beta }})$. A short computation shows
\[ \mathfrak{a}_{\bullet,t} = \mathfrak{a}_\bullet( v_{ (1-t)\boldsymbol {\alpha} + t \boldsymbol {\beta}}) . \]
This computation unfortunately does not generalize to the case of quasi-monomial valuations. See [Reference Liu, Xu and ZhuangLXZ22, § 4] for a study of this failure in the global setting.
To prove Theorem 1.1, we define a measure on 
$\mathbb {R}^2$ that encodes the multiplicities along the geodesic. The argument may be viewed as a local analogue of the construction in [Reference Blum, Liu, Xu and ZhuangBLXZ23, § 3.1] for the geodesic between two filtrations of the section ring of a polarized variety. The latter global construction is motivated by [Reference Boucksom and ChenBC11] and [Reference Boucksom, Hisamoto and JonssonBHJ17, Theorem 4.3], which constructs a measure on 
$\mathbb {R}$ associated to a single filtration of the section ring of polarized variety.
Before proceeding with the proof, fix integers 
$C>0$ and 
$D>0$ such that
\begin{gather*} \mathfrak{m}^{ \lceil C \lambda \rceil } \subset \mathfrak{a}_{\lambda,0} \cap \mathfrak{a}_{\lambda,1},\\ \mathfrak{a}_{D \lambda,0 } \subset \mathfrak{a}_{\lambda, 1} \quad\text{and}\quad \mathfrak{a}_{D\lambda,1} \subset \mathfrak{a}_{\lambda,0}. \end{gather*}
for all 
$\lambda \geq 1$. The existence of 
$C$ follows from Lemma 2.3, while the existence of 
$D$ follows from Lemma 2.3 and the fact that 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ are both linearly bounded by Corollary 3.18.
4.2 Sequences of measures
For each 
$m\in \mathbb {Z}_{>0}$, consider the function 
$H_m\colon \mathbb {R}^2 \to \mathbb {R}$ defined by
\[ H_m(x,y) := \frac{\ell \big( R/ (\mathfrak{a}_{mx,0} \cap \mathfrak{a}_{my,1}) \big)} { m^n/n! } . \]
Notice that 
$H_m$ is non-decreasing and left continuous in each variable. Using the sequence 
$(H_m)_{m\geq 1}$, we define a sequence of measures on 
$\mathbb {R}^2$.
Proposition 4.3 The distributional derivative 
$\mu _m:= - ({ \partial ^2 H_m}/{\partial x \partial y})$ is a discrete measure on 
$\mathbb {R}^2$ and has support contained in 
$\{ ({1}/{D}) x \leq y \leq Dx \} \cup \{ 0 \leq x \leq {1}/{m} \} \cup \{ 0 \leq y\leq {1}/{m} \}$.
Before proving the proposition, we prove the following lemma allowing us to reduce to the case of complete local rings.
Lemma 4.4 Let 
$(\widehat {R},\widehat {\mathfrak {m}})$ denote the completion of 
$(R,\mathfrak {m})$, 
$\mathfrak {b}_{\bullet,i}= \mathfrak {a}_{\bullet,i} \cdot \widehat {R}$ for 
$i=0,1$, and 
$(\mathfrak {b}_{t,\bullet })_{t\in [0,1]}$ the geodesic between 
$\mathfrak {b}_{\bullet,0}$ and 
$\mathfrak {b}_{\bullet,1}$. Then the following statements hold,
(1)
$\ell ( {R}/ \mathfrak {a}_{a, 0} \cap \mathfrak {a}_{b,1})= \ell ( \hat {R}/ \mathfrak {b}_{a, 0} \cap \mathfrak {b}_{b,1})$ for each $a,b\geq 0$.(2)
$\ell ( {R}/ \mathfrak {a}_{a,t}) = \ell ( \hat {R}/ \mathfrak {b}_{a, t} )$ for each $a\geq 0$ and $t\in [0,1]$.
Proof. For 
$\mathfrak {m}$-primary ideals 
$\mathfrak {c}\subset R$ and 
$\mathfrak {d}\subset R$,
\[ (\mathfrak{c} + \mathfrak{d} )\widehat{R} = \mathfrak{c} \widehat{R} + \mathfrak{d} \widehat{R} \quad \text{ and } \quad (\mathfrak{c} \cap \mathfrak{d} )\widehat{R} = \mathfrak{c} \widehat{R} \cap \mathfrak{d} \widehat{R} . \]
Therefore,
\[ (\mathfrak{a}_{a, 0} \cap \mathfrak{a}_{b,1} ) \widehat{R} = \mathfrak{b}_{a, 0} \cap \mathfrak{b}_{b,1} \quad \text{ and } \quad \mathfrak{a}_{a,t} \widehat{R} = \mathfrak{b}_{a, t} . \]
Statements (1) and (2) follow from these equalities.
Proof of Proposition 4.3 By Lemma 4.4, it suffices to prove the statement when 
$(R,\mathfrak {m})$ is complete. Since 
$R$ contains a field by assumption and is complete, there is an inclusion 
$\kappa \hookrightarrow R$ such that the composition 
$\kappa \hookrightarrow R \to \kappa$ is the identity. This will be helpful, since any finite-length 
$R$-module is naturally a finite-dimensional 
$\kappa$-vector space via restriction of scalars.
Fix an integer 
$N>0$ and consider the finite-dimensional 
$\kappa$-vector space 
$V:= R/ \mathfrak {m}^{NCm}$. The 
$\mathfrak {m}$-filtrations 
$\mathfrak {a}_{\bullet,0}$ and 
$\mathfrak {a}_{\bullet,1}$ induce decreasing filtrations 
$\mathcal {F}^\bullet _0$ and 
$\mathcal {F}^\bullet _1$ of 
$V$ defined by
\[ \mathcal{F}_0^\lambda V := {\rm im}( \mathfrak{a}_{\lambda,0} \to V) \quad \text{ and } \quad \mathcal{F}_1^\lambda V := {\rm im}(\mathfrak{a}_{\lambda,1} \to V) . \]
Observe that when 
$x< N$ and 
$y< N$,
\[ \ell \big( R / (\mathfrak{a}_{mx,0} \cap \mathfrak{a}_{my,1}) \big) = \dim_\kappa \big( V / (\mathcal{F}_0^{mx,0} V \cap \mathcal{F}_1^{my,1} V )\big) = \dim_\kappa V- \dim_\kappa (\mathcal{F}_0^{mx,0} V \cap \mathcal{F}_1^{my,1} V), \]
where the first equality uses that 
$\mathfrak {m}^{NCm} \subset \mathfrak {a}_{mx,0} \cap \mathfrak {a}_{my,1}$.
To analyze the dimension of 
$\mathcal {F}_0^{mx} V \cap \mathcal {F}_1^{my} V$, we use that any finite-dimensional vector space with two filtrations admits a basis simultaneously diagonalizing both filtrations (see [Reference Boucksom and ErikssonBE21, Proposition 1.14] and [Reference Abban and ZhuangAZ22, Lemma 3.1]). This means there exists a basis 
$(s_1,\ldots, s_\ell )$ for 
$V$ such that each 
$\mathcal {F}_j^\lambda V$ is the span of some subset of the basis elements. Hence, if we set 
$\lambda _{i,j} := \sup \{ \lambda \in \mathbb {R} \mid s_i \in \mathcal {F}_j^\lambda V \}$, then
\[ \mathcal{F}_0^\lambda V ={\rm span} \langle s_i \mid \lambda_{i,0} \geq\lambda \rangle \quad \text{ and } \quad \mathcal{F}_1^\lambda V ={\rm span} \langle s_i \mid \lambda_{i,1} \geq\lambda \rangle . \]
Using that 
$\dim _\kappa \mathcal {F}_0^{mx} V \cap \mathcal {F}_1^{my} V = \# \{ i \mid \lambda _{i,0} \geq mx \text { and }\lambda _{i,1} \geq my \}$, we compute
\[ \frac{ \partial^2}{\partial x \partial y } \dim_\kappa (\mathcal{F}_0^{mx} V \cap \mathcal{F}_1^{my} V) = \sum_{i=1}^{\ell} \delta_{ (m^{-1} \lambda_{i,0}, m^{-1} \lambda_{i, 1} )} . \]
Therefore, the restriction of 
$\mu _m$ to 
$(-\infty, N)\times (-\infty, N)$ equals
\begin{equation} \mu_m \big\vert_{ (-\infty,N)\times (-\infty,N)} = \frac{1}{m^n/n!} \sum_{ i \mid \lambda_{i,0} , \lambda_{i,1} < mN} \delta_{ (m^{-1} \lambda_{i,0}, m^{-1} \lambda_{i,1} )}. \end{equation}
Since 
$N>0$ was arbitrary, 
$\mu _m$ is a discrete measure on