Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems

Let β>1 be a real number and Tβx=βx(mod1). This paper is concerned with the quantitative recurrence properties of the system ([0,1],Tβ) in some (refined) irregular sets. Specifically, let α1,α2>0 and ψ:N→(0,1) be a positive function; we define the set Eα1,α2β=x∈[0,1):lim infn→∞1nSn(x,β)=α1,lim supn→∞1nSn(x,β)=α2, and calculate the Hausdorff dimension of the set Eα1,α2β(ψ):=x∈Eα1,α2β:|Tβnx−x|<ψ(n)i.m.n∈N, where i.m. stands for infinitely many. Consequently, the Hausdorff dimension of the set E^β(ψ)=x∈[0,1):limn→∞1nSn(x,β)doesnotexist, |Tβnx−x|<ψ(n)i.m.n∈N is also determined.