文摘
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 Uq the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension a5083a6d1" title="Click to view the MathML source">D(q) of Uq for each q∈(1,∞). Furthermore, we prove that the dimension function D:(1,∞)→[0,1] is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in (q′,∞), where 836fe880a27b6ee451e5b0ffadd083c7" title="Click to view the MathML source">q′ denotes the Komornik–Loreti constant: although a58485a9b0c24eebd7dab1d337bc79" title="Click to view the MathML source">D(q)>D(q′) for all a8a35a49" title="Click to view the MathML source">q>q′, we have e6ab4899bac62c111cd20d" title="Click to view the MathML source">D′<0 a.e. in (q′,∞). 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 a8e11268bcd9f69004f6f6c2bbb85f6" title="Click to view the MathML source">U of bases in which b41753bc14e1649565ae7c003167b7a6" title="Click to view the MathML source">x=1 has a unique expansion.