Let 1<β<2 and ε(x,β) be Accessories 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 Padded Overall sizes are investigated from the topological, metric, and dimensional viewpoints.