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

TL;DR

This paper develops a quantitative framework for exact approximation order in Diophantine approximation by introducing the set $E(Ψ,ω)=\{x: x∈W(Ψ), |x−p/q|≥(1−ω(q))Ψ(q) ext{ for large }q\}$ and studying its size under the constraint $ω(q)→0$. The authors prove non-emptiness and uncountability under natural decay and divisibility conditions, and establish precise Hausdorff-dimension formulas; in particular, they relate the dimension to the lower order of $Ψ(q)ω(q)$ via $ ext{dim}_H(E(Ψ,ω))=\inf\{s: \sum_q (Ψ(q)ω(q))^s<∞\}$ under suitable hypotheses, with stronger statements when square divisibility is assumed. They develop a Cantor-type construction to realize large subsets of $E(Ψ,ω)$ and provide specialized results for $Ψ_τ(q)=q^{−τ}$, identifying thresholds for uncountability and dimensions in terms of $ω$-decay rates, including a sharp empty/nonempty dichotomy at $ω(q) \\asymp q^{−τ(τ−1)}$. A key methodological contribution is a detailed analysis of the diophantine window $Ψ(q)(1−ω(q))≤|x−p/q|≤Ψ(q)$, which is encoded symbolically via continued fractions and leveraged to control extraneous rational approximations. Overall, the work sharpens the understanding of when $Ψ$ is the optimal approximation function for $x$, linking metric Diophantine properties to Cantor-structure arguments and continued-fraction dynamics with explicit threshold phenomena and dimension formulas.

Abstract

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

Theorems & Definitions (62)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10