A quantitative framework for sets of exact approximation order by rational numbers

TL;DR

This paper introduces a new quantitative framework in Diophantine approximation to analyze the size and structure of sets of points with specific approximation properties, revealing thresholds for their cardinality and Hausdorff dimension.

Contribution

It develops a novel approach to measure exact approximation order by rational numbers, establishing conditions for non-emptiness, uncountability, and dimension of approximation sets.

Findings

Existence of approximation sets with prescribed order and non-emptiness.

Conditions under which these sets are uncountable and have positive Hausdorff dimension.

Identification of thresholds where these sets become empty.

Abstract

In this paper we study a quantitative notion of exactness within Diophantine approximation. Given $\Psi:(0,\infty)\to (0,\infty)$ and $\omega:(0,\infty)\to (0,1)$ satisfying $\lim_{q\to\infty}\omega(q)=0$, we study the set of points, which we call $E(\Psi,\omega)$, that are $\Psi$-well approximable but not $\Psi(1-\omega)$-well approximable. We prove results on the cardinality and dimension of $E(\Psi,\omega)$. In particular we obtain the following general statements: (i) For any $\omega:(0,\infty)\to (0,1)$ and $\tau>2$ there exists $\Psi:(0,\infty)\to (0,\infty)$ such that $\lim_{q\to\infty}\frac{-\log \Psi(q)}{\log q}=\tau$ and $E(\Psi,\omega)\neq\emptyset.$ (ii) Under natural monotonicity assumptions on $\Psi$ and $\omega,$ we prove that if $\omega$ decays to zero sufficiently slowly (in a way that depends upon $\Psi$) then $E(\Psi,\omega)$ is uncountable. Moreover, under further natural assumptions on $\Psi$ we can calculate the Hausdorff dimension of $E(\Psi,\omega)$. Our main result demonstrates a new threshold for the behaviour of $E(\Psi,\omega)$. A particular instance of this threshold is illustrated by considering functions of the form $\Psi_{\tau}(q)=q^{-\tau}$ when $\tau\in \mathbb{N}_{\geq 3}$. For these functions we prove the following: (iii) If $\omega(q)= Cq^{-\tau(\tau-1)}$ for some sufficiently large $C$ or $\omega(q)=q^{-\tau'}$ for some $\tau'<\tau(\tau-1),$ then $E(\Psi_{\tau},\omega)$ is uncountable and we calculate its Hausdorff dimension. (iv) If $\omega(q)< cq^{-\tau(\tau-1)}$ for some $c\in (0,1)$ for all $q$ sufficiently large then $E(\Psi_{\tau},\omega)=\emptyset.$