Abstract. We fix a positive integer M , and we consider expansions in arbitrary real bases q > 1 over the alphabet {0, 1, . . . , M }. We denote by U q the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension D(q) of U q for each q ∈ (1, ∞). Furthermore, we prove that the dimension function D : (1, ∞) → [0, 1] is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in (q ′ , ∞), where. During the proofs we improve and generalize a theorem of Erdős et al. on the existence of large blocks of zeros in β-expansions, and we determine for all M the Lebesgue measure and the Hausdorff dimension of the set U of bases in which x = 1 has a unique expansion.

Abstract. Given an integer N ≥ 2 and a real number β > 1, let Γ β,N be the set of allwhere β L is a purely Parry number while β U is a transcendental number whose quasi-greedy expansion of 1 is related to the classical Thue-Morse sequence.This allows us to calculate the Hausdorff dimension of U β,N for almost every β > 1. In particular, this improves the main results of Gábor Kallós (1999Kallós ( , 2001). Moreover, we find that the dimension function f (β) = dim H U β,N fluctuates frequently for β ∈ (1, N ).

Let Γ β,N be the N -part homogeneous Cantor set with β ∈ (1/(2N − 1), 1/N ).is called a code of t. Let U β,±N be the set of t ∈ [−1, 1] having a unique code, and let S β,±N be the set of t ∈ U β,±N which make the intersection Γ β,N ∩ (Γ β,N + t) a selfsimilar set. We characterize the set U β,±N in a geometrical and algebraical way, and give a sufficient and necessary condition for t ∈ S β,±N . Using techniques from beta-expansions, we show that there is a critical point βc ∈ (1/(2N − 1), 1/N ), which is a transcendental number, such that U β,±N has positive Hausdorff dimension if β ∈ (1/ (2N − 1), βc), and contains countably infinite many elements if β ∈ (βc, 1/N ). Moreover, there exists a second critical point αc = N + 1 − (N − 1)(N + 3) /2 ∈ (1/ (2N − 1), βc) such that S β,±N has positive Hausdorff dimension if β ∈ (1/ (2N − 1), αc), and contains countably infinite many elements if β ∈ [αc, 1/N ).

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$ . We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$ , where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.