Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems

Let β > 1 be a real number and T β x = β x ( m o d 1 ) . This paper is concerned with the quantitative recurrence properties of the system ( [ 0 , 1 ] , T β ) in some (refined) irregular sets. Specifically, let α 1 , α 2 > 0 and ψ : N → ( 0 , 1 ) be a positive function; we define the set E α 1 , α 2 β = x ∈ [ 0 , 1 ) : lim inf n → ∞ 1 n S n ( x , β ) = α 1 , lim sup n → ∞ 1 n S n ( x , β ) = α 2 , and calculate the Hausdorff dimension of the set E α 1 , α 2 β ( ψ ) : = x ∈ E α 1 , α 2 β : | T β n x − 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 ) : lim n → ∞ 1 n S n ( x , β ) does not exist , | T β n x − x | < ψ ( n ) i . m . n ∈ N is also determined.