arXiv subjects

Jason Levesley

Publications and source records attributed to Jason Levesley.

10 recordsLinked to original sources

arXiv2024

Shrinking targets versus recurrence: the quantitative theory

Let $X = [0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $\psi:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[ R(x,N;T,\psi) := \#\{1\leq n\leq N: d(T^n x, x) < \psi(n)\}. \] We show that for any $\varepsilon > 0$ we have \[ R(x,N;T,\psi) = \Psi(N)+O\left(\Psi^{1/2}(N) \ (\log\Psi(N))^{3/2+\varepsilon}\right) \] for $\mu$-almost all $x\in X$ and for all $N\in\mathbb N$, where $\Psi(N):= 2 \sum_{n=1}^N \psi(n)$. We also prove a generalization of this result to higher dimensions.

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arXiv2023

The dimension of the set of $ψ$-badly approximable points in all ambient dimensions; on a question of Beresnevich and Velani

Let $ψ:\mathbb{N} \to [0,\infty)$, $ψ(q)=q^{-(1+τ)}$ and let $ψ$-badly approximable points be those vectors in $\mathbb{R}^{d}$ that are $ψ$-well approximable, but not $cψ$-well approximable for arbitrarily small constants $c>0$. We establish that the $ψ$-badly approximable points have the Hausdorff dimension of the $ψ$-well approximable points, the dimension taking the value $(d+1)/(τ+1)$ familiar from theorems of Besicovitch and Jarník. The method of proof is an entirely new take on the Mass Transference Principle by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named `delayed pruning' to construct a sufficiently large $\liminf$ set and combine this with ideas inspired by the proof of the Mass Transference Principle to find a large $\limsup$ subset of the $\liminf$ set. Our results are a generalisation of some $1$-dimensional results due to Bugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothing alike.

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arXiv2021

Simultaneous p-adic Diophantine approximation

The goal 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, and a Zero-One law for weighted $p$-adic approximations established in this paper.

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arXiv2020

A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds

Given a weight vector $τ=(τ_{1}, \dots, τ_{n}) \in \mathbb{R}^{n}_{+}$ with each $τ_{i}$ bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set of $τ$-approximable points points over a manifold $\mathcal{M}$, where $\mathcal{M}$ is twice continuously differentiable. From this we produce a lower bound for the set of $ψ$-approximable points over a manifold where $ψ$ is a general approximation function with certain limits. The proof is based on a technique developed by Beresnevich et al. in arXiv:1712.03761, but we use an alternative mass transference style theorem.

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arXiv2015

Diophantine Approximation and applications in Interference Alignment

This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of $\mathbb{R}^n$. The importance of such quantitative statements is explicitly discussed in Section 4.7.1 of Jafar's monograph `Interference Alignment - A New Look at Signal Dimensions in a Communication Network', Foundations and Trends in Communications and Information Theory, Vol. 7, no. 1, 2010.

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arXiv2013

A converse to linear independence criteria, valid almost everywhere

We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational numbers.

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arXiv2010

The mixed Schmidt conjecture in the theory of Diophantine approximation

Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) > 0$ such that \[ \max\{|q|_\DDD^{1/i}, \|qx\|^{1/j}\} > c(x)/ q \qquad \forall q \in \NN \] is one quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. In turn, this establishes the natural analogue of Schmidt's conjecture within the framework of the de Mathan-Teulié conjecture -- also known as the `Mixed Littlewood Conjecture'.

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arXiv2009

The Metrical Theory of Simultaneously Small Linear Forms

In this paper we investigate the metrical theory of Diophantine approximation associated with linear forms that are simultaneously small for infinitely many integer vectors; i.e. forms which are close to the origin. A complete Khintchine--Groshev type theorem is established, as well as its Hausdorff measure generalization. The latter implies the complete Hausdorff dimension theory.

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arXiv2006

Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves

Let $\mathcal{C}$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation with two independent approximation functions; that is if a certain sum converges then the set of all points $(x,y)$ on the curve which satisfy simultaneously the inequalities $\| q x \| < ψ_1(q)$ and $\| qy \| < ψ_2(q)$ infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for the cae of multiplicative approximation $\| qx \| \| q y \| < ψ(q)$, we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

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arXiv2005

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2, >... \} $. Given a real, positive function $ψ$ let $ W_{\cal A}(ψ)$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely many $(p,q) \in \Z \times {\cal A} $ such that $ |x - p/q| < ψ(q) $. The analogue of the Hausdorff measure version of the Duffin-Schaeffer conjecture is established for $ W_{\cal A}(ψ) \cap K $. One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in $K$ -- an assertion attributed to K. Mahler.

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