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

MAKE IMPACT

Simon Baker , Benjamin Ward · YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2510.18451

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 $E(Ψ_τ,ω)=\emptyset.$

Type

preprint

Landing Page

http://arxiv.org/abs/2510.18451

PDF

https://arxiv.org/pdf/2510.18451

OA Status

green

OpenAlex ID

https://openalex.org/W4416056109

Raw OpenAlex JSON

OpenAlex ID

https://openalex.org/W4416056109

Canonical identifier for this work in OpenAlex

DOI

https://doi.org/10.48550/arxiv.2510.18451

Digital Object Identifier

Title

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

Work title

Type

preprint

OpenAlex work type

Publication year

2025

Year of publication

Publication date

2025-10-21

Full publication date if available

Authors

Simon Baker, Benjamin Ward

List of authors in order

Landing page

https://arxiv.org/abs/2510.18451

Publisher landing page

PDF URL

https://arxiv.org/pdf/2510.18451

Direct link to full text PDF

Open access

Yes

Whether a free full text is available

OA status

green

Open access status per OpenAlex

OA URL

https://arxiv.org/pdf/2510.18451

Direct OA link when available

Cited by

0

Total citation count in OpenAlex

Full payload