Univoque bases and Hausdorff dimension.

Given a positive integer M and a real number $$q >1$$ , a q -expansion of a real number x is a sequence $$(c_i)=c_1c_2\ldots $$ with $$(c_i) \in \{0,\ldots ,M\}^\infty $$ such that It is well known that if $$q \in (1,M+1]$$ , then each $$x \in I_q:=\left[ 0,M/(q-1)\right] $$ has a q-expansion. Let $$\mathcal {U}=\mathcal {U}(M)$$ be the set of univoque bases $$q>1$$ for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of $$\mathcal {U}$$ and to show that the Hausdorff dimension of the set of numbers $$x \in I_q$$ with a unique q-expansion changes the most if q 'crosses' a univoque base. Denote by $$\mathcal {B}_2=\mathcal {B}_2(M)$$ the set of $$q \in (1,M+1]$$ such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741-754, 2009) and prove that where $$q'=q'(M)$$ is the Komornik-Loreti constant. [ABSTRACT FROM AUTHOR]