Quantitative recurrence properties in Besicovitch-Eggleston sets

Abstract

Let Tx=bx(mod1) and Ax(i,b,N) denote the number of occurrences of the digit i in the first N digits of b-adic expansion of x. We study the quantitative recurrence properties in Besicovitch-Eggleston sets, which is defined asRb,ρ(ψ)={x∈[0,1):|Tnx−x|<ψ(n)for infinitely manyn∈NandlimN→∞⁡Ax(i,b,N)N=pifor anyi∈{0,⋯,b−1}}, where ρ=(p0,⋯,pb−1) is a probability vector and ψ: N→(0,1] is a positive function. In this paper, the Hausdorff dimension of the set Rb,ρ(ψ) is determined.