Finite and Infinite Arithmetic Progressions Related to Beta-Expansion

Finite and Infinite Arithmetic Progressions Related to Beta-Expansion

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Sunglasses 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, Queen Bookcase Platform Bed metric, and dimensional viewpoints.