Uniform Diophantine approximation with restricted denominators

Abstract

Let b≥2 be an integer and A=(an)n=1∞ be a strictly increasing subsequence of positive integers with η:=lim supn→∞an+1an<+∞. For each irrational real number ξ, we denote by vˆb,A(ξ) the supremum of the real numbers vˆ for which, for every sufficiently large integer N, the equation ‖banξ‖<(baN)−vˆ has a solution n with 1≤n≤N. For every vˆ∈[0,η], let Vˆb,A(vˆ) (Vˆb,A⁎(vˆ)) be the set of all real numbers ξ such that vˆb,A(ξ)≥vˆ (vˆb,A(ξ)=vˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ). When η=1, we prove that the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) are equal to (1−vˆ1+vˆ)2 for any vˆ∈[0,1]. When η>1 and limn→∞⁡an+1an exists, we show that the Hausdorfff dimension of Vˆb,A(vˆ) is strictly less than (η−vˆη+vˆ)2 for some vˆ, which is different with the case η=1, and we give a lower bound of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) for any vˆ∈[0,η]. Furthermore, we show that this lower bound can be reached for some vˆ.