Finite and Infinite Arithmetic Progressions Related to Beta-Expansion

and Applied Analysis 3 Proposition 6. The set E is of the first category. Proof. The set E can be written as E = ∞ ⋂ k=1 ∞ ⋃ N=1 ∞ ⋂ n=N {x ∈ [0, 1) : Nn (x, β) n < 1


Introduction
Let  be a sequence of natural numbers.The lower asymptotic density () of  (the upper asymptotic density () of , resp.) is defined as If () = (), the common value is called the asymptotic density of , denoted by ().
The concept of asymptotic density was extensively applied to number theory such as arithmetic progressions, the first of which is the following theorem due to Szemerédi [1].
The issue to find what kind of sequence  can contain arbitrarily long arithmetic progressions is very popular nowadays.For instance, Green and Tao [2] proved that the sequence  of prime numbers contains arbitrarily long arithmetic progressions, which extends Theorem 1 since () = 0.
Assume that the sequence  contains arbitrarily long arithmetic progressions; a further question is whether  can contain an infinite arithmetic progression.In 1972, Wagstaff Jr. [3] found a sequence of natural numbers which contains arbitrarily long arithmetic progressions but does not have any infinite one.In fact, he even gave more information as the following.
The topics of arithmetic progressions and binary expansion were firstly linked by Šalát and Tomanová [4].They considered the sequence of the positions where the digit 1 appears in the binary expansion of real numbers in the unit interval.In this paper, we investigate the same questions for -expansion with 1 <  ≤ 2 and generalize the results in [4].
Obviously, I  ⊂ F  since an infinite arithmetical progression contains arbitrarily long ones.In Section 2, we will give some introduction to -expansion.In Sections 3 and 4, we will describe the sets F  , I  and their complements F   , I   from the viewpoints of topology, metric, and Hausdorff dimension.
When  = 2, the Lebesgue measure L is a   -invariant measure.If  is not an integer, the Lebesgue measure L is not   -invariant; however Rényi [7] proved that there exists a unique invariant measure   equivalent to L. Parry [5] and Gel'fond [8] independently gave the density formula of   and L.
So we complete the proof.
Secondly, combining the facts that and the set  is of the first category (Proposition 6), we know that the set F   is of the first category.So the set F  is residual.

Metric and Dimensional Properties of
)) > 0. (20) Note that   is equivalent to the Lebesgue measure L; then, for L-a.e. , the asymptotic density of   () Thus L(F  ) = 1.
⊂ I