Finite and Infinite Arithmetic Progressions Related to Beta-Expansion

Abstract EN

Let 1<β<2 and ε(x,β) be the β-expansion of x∈[0,1).

Denote by Aβ(x) the set of positions where the digit 1 appears in ε(x,β).

We consider the sets of points x such that Aβ(x) contains arbitrarily long arithmetic progressions and includes infinite arithmetic progressions, respectively.

Their sizes are investigated from the topological, metric, and dimensional viewpoints.