文摘
We fix a positive integer M , and we consider expansions in arbitrary real bases k to view the MathML source">q>1den">de"> over the alphabet k to view the MathML source">{0,1,…,M}den">de">. We denote by dee874ea4da3d9" title="Click to view the MathML source">Uqden">de"> the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension k to view the MathML source">D(q)den">de"> of dee874ea4da3d9" title="Click to view the MathML source">Uqden">de"> for each e8d9bf05dfe8" title="Click to view the MathML source">q∈(1,∞)den">de">. Furthermore, we prove that the dimension function k to view the MathML source">D:(1,∞)→[0,1]den">de"> is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in k to view the MathML source">(q′,∞)den">de">, where e880a27b6ee451e5b0ffadd083c7" title="Click to view the MathML source">q′den">de"> denotes the Komornik–Loreti constant: although k to view the MathML source">D(q)>D(q′)den">de"> for all k to view the MathML source">q>q′den">de">, we have k to view the MathML source">D′<0den">de"> a.e. in k to view the MathML source">(q′,∞)den">de">. 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 k to view the MathML source">Uden">de"> of bases in which k to view the MathML source">x=1den">de"> has a unique expansion.