A quantitative framework for sets of exact approximation order by rational numbers
Authors
Simon Baker
Benjamin Ward
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 $E(Ψ_τ,ω)=\emptyset.$
