No abstract

Expansion of real number is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we systematically begin to study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy, lazy, quasi-lazy and unique expansions in multiple bases.Note that a 0 = 0 and b m = m βm−1 . For all m ∈ N, let

We obtain an exact formula of the Hausdorff dimension of the level sets in beta-expansions for pseudo-golden ratios by using a variation formula. Before this, we prove that the Hausdorff dimension of an arbitrary set in the shift space is equal to its projection in [0, 1], and we clarify that for calculating the Hausdorff dimension of the level sets, one only needs to focus on the Markov measures when β ∈ (1, 2) and the beta-expansion of 1 is finite.

The structures of full words and non-full for β-expansions are completely characterized in this paper. We obtain the precise lengths of all the maximal runs of full and non-full words among admissible words with same order.