We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure coreplatform@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Let $\psi : \mathbb {N} \to [0,1/2]$ be given. The Duffin–Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|\alpha - p/q| < \psi (q)/q$, provided that the series $\sum _{q=1}^\infty \varphi (q) \psi (q) / q$ is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all $\alpha$ the number of coprime solutions $(p,q)$, subject to $q \leq Q$, is of asymptotic order $\sum _{q=1}^Q 2 \varphi (q) \psi (q) / q$. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate from sieve theory, and number-theoretic input on the ‘anatomy of integers’. The key phenomenon is that the system of approximation sets exhibits ‘asymptotic independence on average’ as the total mass of the set system increases.
We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers
$x \in (0,1]$
with the following property is comeager: for all integers
$b\ge 2$
and
$k\ge 1$
, the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.
Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.
We show that for a Salem number
$\beta $
of degree d, there exists a positive constant
$c(d)$
where
$\beta ^m$
is a Parry number for integers m of natural density
$\ge c(d)$
. Further, we show
$c(6)>1/2$
and discuss a relation to the discretized rotation in dimension
$4$
.
Let
$r=[a_1(r), a_2(r),\ldots ]$
be the continued fraction expansion of a real number
$r\in \mathbb R$
. The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let
$(t_1, \ldots , t_m)\in \mathbb R_+^m$
, and let
$\Psi :\mathbb {N}\rightarrow (1,\infty )$
be a function such that
$\Psi (n)\to \infty $
as
$n\to \infty $
. We calculate the Hausdorff dimension of the set of all
$ (x, y)\in [0,1)^2$
such that
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.
Let
$0\leq \alpha \leq \infty $
,
$0\leq a\leq b\leq \infty $
and
$\psi $
be a positive function defined on
$(0,\infty )$
. This paper is concerned with the growth of
$L_{n}(x)$
, the largest digit of the first n terms in the Lüroth expansion of
$x\in (0,1]$
. Under some suitable assumptions on the function
$\psi $
, we completely determine the Hausdorff dimensions of the sets
We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and
$\alpha $
-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret’s tail-equivalence theorem for almost all points. The results are new even in the case of some real and complex continued fractions.
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.
Given any rectangular polyhedron
$3$
-manifold
$\mathcal {P}$
tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in
$\mathcal {P}$
.
In this paper, we discuss a connection between geometric measure theory and number theory. This method brings a new point of view for some number-theoretic problems concerning digit expansions. Among other results, we show that for each integer k, there is a number
$M>0$
such that if
$b_{1},\ldots ,b_{k}$
are multiplicatively independent integers greater than M, there are infinitely many integers whose base
$b_{1},b_{2},\ldots ,b_{k}$
expansions all do not have any zero digits.
We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.
The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.
For any x in
$[0,1)$
, let
$[a_1(x),a_2(x),a_3(x),\ldots ]$
be its continued fraction. Let
$\psi :\mathbb {N}\to \mathbb {R}^+$
be such that
$\psi (n) \to \infty $
as
$n\to \infty $
. For any positive integers s and t, we study the 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.
A Cantor series expansion for a real number x with respect to a basic sequence
$Q=(q_1,q_2,\dots )$
, where
$q_i \geq 2$
, is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a
$\boldsymbol {\Pi }^0_3$
-complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is
$\boldsymbol {\Pi }^0_3$
-complete, and if Q is
$1$
-divergent then the sets of normal and ratio normal numbers are
$\boldsymbol {\Pi }^0_3$
-complete. We further show that all five non-trivial differences of these sets are
$D_2(\boldsymbol {\Pi }^0_3)$
-complete if
$\lim _i q_i=\infty $
and Q is
$1$
-divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible.
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.
We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$.
We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.
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.