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In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$, the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$. Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$.
We consider the set of elements in a translation of the middle-third Cantor set which can be well approximated by algebraic numbers of bounded degree. A doubling dimensional result is given, which enables one to conclude an upper bound on the dimension of the set in question for a generic translation.
Motivated by the p-adic approach in two of Mahler’s problems, we obtain some results on p-adic analytic interpolation of sequences of integers
$(u_n)_{n\geq 0}$
. We show that if
$(u_n)_{n\geq 0}$
is a sequence of integers with
$u_n = O(n)$
which can be p-adically interpolated by an analytic function
$f:\mathbb {Z}_p\rightarrow \mathbb {Q}_p$
, then
$f(x)$
is a polynomial function of degree at most one. The case
$u_n=O(n^d)$
with
$d>1$
is also considered with additional conditions. Moreover, if X and Y are subsets of
$\mathbb {Z}$
dense in
$\mathbb {Z}_p$
, we prove that there are uncountably many p-adic analytic injective functions
$f:\mathbb {Z}_p\to \mathbb {Q}_p$
, with rational coefficients, such that
$f(X)=Y$
.
We prove that for any infinite sets of nonnegative integers $\mathcal {A}$ and $\mathcal {B}$, there exist transcendental analytic functions $f\in \mathbb {Z}\{z\}$ whose coefficients vanish for any indexes $n\not \in \mathcal {A}+\mathcal {B}$ and for which $f(z)$ is algebraic whenever z is algebraic and $|z|<1$. As a consequence, we provide an affirmative answer for an asymptotic version of Mahler’s problem A.
In this paper, we investigate pigeonhole statistics for the fractional parts of the sequence $\sqrt {n}$. Namely, we partition the unit circle $ \mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence $(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as $N \to \infty $. More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$. A natural way to examine this is via point processes—random measures on $[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in $\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J.123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.
Dirichlet’s theorem, including the uniform setting and asymptotic setting, is one of the most fundamental results in Diophantine approximation. The improvement of the asymptotic setting leads to the well-approximable set (in words of continued fractions)
$$ \begin{align*} \mathcal{K}(\Phi):=\{x:a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}; \end{align*} $$
the improvement of the uniform setting leads to the Dirichlet non-improvable set
$$ \begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*} $$
Surprisingly, as a proper subset of Dirichlet non-improvable set, the well-approximable set has the same s-Hausdorff measure as the Dirichlet non-improvable set. Nevertheless, one can imagine that these two sets should be very different from each other. Therefore, this paper is aimed at a detailed analysis on how the growth speed of the product of two-termed partial quotients affects the Hausdorff dimension compared with that of single-termed partial quotients. More precisely, let
$\Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$
be two non-decreasing positive functions. We focus on the Hausdorff dimension of the set
$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$
. It is known that the dimensions of
$\mathcal {G}(\Phi )$
and
$\mathcal {K}(\Phi )$
depend only on the growth exponent of
$\Phi $
. However, rather different from the current knowledge, it will be seen in some cases that the dimension of
$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$
will change greatly even slightly modifying
$\Phi _1$
by a constant.
We build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of
$\mathrm {K}$
-semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the Harder and Narasimhan polygons. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138). One source of inspiration for our abstract study of Harder and Narasimhan data, which is a concept that we define here, is the lattice reduction methods of Grayson (1984, Commentarii Mathematici Helvetic 59, 600–634). Another is the work of Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138), and Evertse and Ferretti (2013, Annals of Mathematics 177, 513–590), which is within the context of Diophantine approximation for projective varieties.
We consider the two-dimensional shrinking target problem in beta dynamical systems (for general
$\beta>1$
) with general errors of approximation. Let
$f, g$
be two positive continuous functions. For any
$x_0,y_0\in [0,1]$
, define the shrinking target set
where
$S_nf(x)=\sum _{j=0}^{n-1}f(T_\beta ^jx)$
is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher-dimensional beta dynamical systems.
For an (irreducible) recurrence equation with coefficients from
$\mathbb Z[n]$
and its two linearly independent rational solutions
$u_n,v_n$
, the limit of
$u_n/v_n$
as
$n\to \infty $
, when it exists, is called the Apéry limit. We give a construction that realises certain quotients of L-values of elliptic curves as Apéry limits.
In this paper, we consider an equivalence relation on the space
$AP(\mathbb {R},X)$
of almost periodic functions with values in a prefixed Banach space X. In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove that every equivalence class is sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class, i.e., the condition of almost periodicity of a function
$f\in AP(\mathbb {R},X)$
yields that every sequence of translates of f has a subsequence that converges to a function equivalent to f. This extends previous work by the same authors on the case of numerical almost periodic functions.
We define certain arithmetic derivatives on
$\mathbb {Z}$
that respect the Leibniz rule, are additive for a chosen equation
$a+b=c$
, and satisfy a suitable nondegeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the
$abc$
Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the
$abc$
Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighties, relating Geometry of Numbers in arithmetic to derivatives in function fields and Nevanlinna theory. In addition, our construction formalizes the widespread intuition that the
$abc$
Conjecture should be related to arithmetic derivatives of some sort.
We show that geodesics in
$\mathbf {H}$
attached to a maximal split torus or a real quadratic torus in
$GL_{2, \mathbf {Q}}$
are the only irreducible algebraic curves in
$\mathbf {H}$
whose image in
$\mathbf {R}^2$
via the j-invariant is contained in an algebraic curve.
A celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.
This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number
$m,$
we determine the Hausdorff dimension of the following set:
where
$\tau $
is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when
$m=1$
) shown by Hussain, Kleinbock, Wadleigh and Wang.
In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of
$U_{m}$
-numbers by using continued fraction expansions of algebraic formal power series of degree
$m>1$
.
Given an infinite subset
$\mathcal{A} \subseteq\mathbb{N}$
, let A denote its smallest N elements. There is a rich and growing literature on the question of whether for typical
$\alpha\in[0,1]$
, the pair correlations of the set
$\alpha A (\textrm{mod}\ 1)\subset [0,1]$
are asymptotically Poissonian as N increases. We define an inhomogeneous generalisation of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.
An effective estimate for the measure of the set of real numbers for which the inequality |P(x)|<Q-w for $w > {3 \over 2}n + 1$ has a solution in integral polynomials P of degree n and of height H(P) at most $Q \in {\rm{\mathbb N}}$ is obtained.
Let q a prime power and
${\mathbb F}_q$
the finite field of q elements. We study the analogues of Mahler’s and Koksma’s classifications of complex numbers for power series in
${\mathbb F}_q((T^{-1}))$
. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.
We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.