The Generalised Baker–Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a non-degenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary non-degenerate manifolds. The divergence part has also been resolved for the $p$-adic setting by Datta-Ghosh in 2022, for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open question. In this paper, we prove the homogeneous $p$-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.

We strengthen known results on Diophantine approximation with restricted denominators by presenting a new quantitative Schmidt-type theorem that applies to denominators growing much more slowly than in previous works. In particular, we can handle sequences of denominators with polynomial growth and Rajchmann measures exhibiting arbitrary slow decay, allowing several applications. For instance, our results yield non-trivial lower bounds on the Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers (each with restricted denominators) and enable the construction of Salem subsets of well-approximable numbers of arbitrary Hausdorff dimension.

In this article, we extend, with a great deal of generality, many results regarding the Hausdorff dimension of certain dynamical Diophantine coverings and shrinking target sets associated with a conformal iterated function system (IFS) previously established under the so-called open set condition. The novelty of the result we present is that it holds regardless of any separation assumption on the underlying IFS and thus extends to a large class of IFSs the previous results obtained by Beresnevitch and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992] and by Barral and Seuret [The multifractal nature of heterogeneous sums of Dirac masses. Math. Proc. Cambridge Philos. Soc. 144(3) (2008), 707–727]. Moreover, it will be established that if S is conformal and satisfies mild separation assumptions (which are, for instance, satisfied for any self-similar IFS on $\mathbb {R}$ with algebraic parameters, no exact overlaps and similarity dimension smaller than $1$), then the classical result of Hill–Velani regarding the shrinking target problem associated with a conformal IFS satisfying the open set condition (and for which the Hausdorff measure was later computed by Allen and Barany [On the Hausdorff measure of shrinking target sets on self-conformal sets. Mathematika 67 (2021), 807–839]) can be extended.

Let $b \geqslant 3$ be an integer and C(b, D) be the set of real numbers in [0,1] whose base b expansion only consists of digits in a set $D {\subseteq} \{0,...,b-1\}$. We study how close can numbers in C(b, D) be approximated by rational numbers with denominators being powers of some integer t and obtain a zero-full law for its Hausdorff measure in several circumstances. When b and t are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann. 338 (2007), 97–118) and generalise their theorem. When b and t are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set

$$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$

$y\in [0,1)$

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and $1$ Cantor winning in metric spaces, and the fact that $1/2$ winning implies absolute winning for subsets of $\mathbb {R}$. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3) 101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $-approximable points and for the set of inhomogeneous multiplicative $\psi $-approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math. 740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$, to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

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.

The aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on p-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.

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.

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)

$$ \begin{align*} \mathcal{K}(\Phi):=\{x:a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}; \end{align*} $$

$$ \begin{align*} \mathcal{G}(\Phi):=\{x:a_{n}(x)a_{n+1}(x)\ge\Phi(q_{n}(x))\ \textrm{for infinitely many }n\in \mathbb{N}\}. \end{align*} $$

$\Phi _{1},\Phi _{2}:[1,+\infty )\rightarrow \mathbb {R}^{+}$

$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$

$\mathcal {G}(\Phi )$

$\mathcal {K}(\Phi )$

$\Phi $

$\mathcal {G}(\Phi _{1})\!\setminus\! \mathcal {K}(\Phi _{2})$

$\Phi _1$

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

$$ \begin{align*}E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{@{}ll@{}} \lvert T_{\beta}^{n}x-x_{0}\rvert <e^{-S_nf(x)}\\[1ex] \lvert T_{\beta}^{n}y-y_{0}\rvert < e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \mathbb N \right\}, \end{align*} $$

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 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 \[||x||\] denote the distance from \[x \in \mathbb{R}\] to the nearest integer. In this paper, we prove a new existence and density result for matrices \[A \in {\mathbb{R}^{m \times n}}\] satisfying the inequality

\[\mathop {\lim \inf }\limits_{|q{|_\infty } \to+ \infty } \prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} \log {\left( {\prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} } \right)^{m + n - 1}}\prod\limits_{i = 1}^m {{A_i}q}> 0,\]

where q ranges in \[{\mathbb{Z}^n}\] and Ai denote the rows of the matrix A. This result extends previous work of Moshchevitin both to arbitrary dimension and to the inhomogeneous setting. The estimates needed to apply Moshchevitin’s method to the case m > 2 are not currently available. We therefore develop a substantially different method, based on Cantor-like set constructions of Badziahin and Velani. Matrices with the above property also appear to have very small sums of reciprocals of fractional parts. This fact helps us to shed light on a question raised by Lê and Vaaler on such sums, thereby proving some new estimates in higher dimension.

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$,

$$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$

$n\in \mathbb {N}$

$\tau $

$a_{n}(x)$

$r,\tau (x)$

$h(x)$

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math. 138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math. 10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math. 24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

For an irrational number $x\in [0,1)$, let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x):n\geq 1\}$. Given $\unicode[STIX]{x1D6E9}\in \mathbb{N}$, for $n\geq 1$, the $n$th longest block function of $x$ with respect to $\unicode[STIX]{x1D6E9}$ is defined by $L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$, which represents the length of the longest consecutive sequence whose elements are all $\unicode[STIX]{x1D6E9}$ from the first $n$ partial quotients of $x$. We consider the growth rate of $L_{n}(x,\unicode[STIX]{x1D6E9})$ as $n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.