Multiple expansions of real numbers with digits set $$\left\{ 0,1,q\right\} $$ 0 , 1 , q

Abstract

For q > 1 we consider expansions in base q with digits set {0, 1, q}. Let U q be the set of points which have a unique q-expansion. For k = 2, 3, . . . , ℵ 0 let B k be the set of bases q > 1 for which there exists x having precisely k different q-expansions, and for q ∈ B k letq be the set of all such x's which have exactly k different q-expansions. In this paper we show thatwhere q c ≈ 2.32472 is the appropriate root of x 3 − 3x 2 + 2x − 1 = 0. Moreover, we show that for any integer k ≥ 2 and any q ∈ B k the Hausdorff dimensions of U (k) q and U q are the same, i.e.,Finally, we conclude that the set of points having a continuum of q-expansions has full Hausdorff dimension.

On the structure of λ-Cantor set with overlaps

“…Given q > 1, Dajani, Kan, Kong and Li in [5] considered expansions in base q with digits set {0, 1, q}. They described the size of sets of points having finite q-expansions.…”