2025/10/21 by Simon Baker, Baker, Simon, Benjamin Ward +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2510.18451
openalex publication_date 2025/10/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
In this paper we study a quantitative notion of exactness within Diophantine approximation. Given Ψ:(0,∞)→ (0,∞) and ω:(0,∞)→ (0,1) satisfying limq→∞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,∞)→ (0,1) and τ>2 there exists Ψ:(0,∞)→ (0,∞) such that limq→∞(-log Ψ(q))/(log q)=τ and E(Ψ,ω)≠∅. (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 Ψτ(q)=q-τ when τ∈ ℕ≥ 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(Ψτ,ω) is uncountable and we calculate its Hausdorff dimension. (iv) If ω(q)< cq-τ(τ-1) for some c∈ (0,1) for all q sufficiently large then E(Ψτ,ω)=∅.