Multiplicative Diophantine approximation with restricted denominators

2024/09/27 by Bing Li, Ruofan Li, Li, Bing +3
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Algebraic Geometry and Number Theory #Advanced Mathematical Theories and Applications

paper · pdf · doi:10.48550/arxiv.2409.18635

Abstract

Let \a_n\_n∈ℕ, \b_n\_{n∈ ℕ} be two infinite subsets of positive integers and ψ:ℕ→ ℝ_>0 be a positive function. We completely determine the Hausdorff dimensions of the set of all points (x,y)∈ [0,1]² which satisfy ‖a_nx‖‖b_ny‖<ψ(n) infinitely often, and the set of all x∈ [0,1] satisfying ‖a_nx‖‖b_nx‖<ψ(n) infinitely often. This is based on establishing general convergence results for Hausdorff measures of these two sets. We also obtain some results on the set of all x∈ [0,1] such that max\‖a_nx‖, ‖b_nx‖\<ψ(n) infinitely often.

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