文摘
We fix a positive integer M , and we consider expansions in arbitrary real bases q>1 over the alphabet b1668e7ed06dd978d" title="Click to view the MathML source">{0,1,…,M}. We denote by b193b0c769459ba20dee874ea4da3d9" title="Click to view the MathML source">Uq the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension D(q) of b193b0c769459ba20dee874ea4da3d9" title="Click to view the MathML source">Uq for each 822e516c827a07fce8d9bf05dfe8" title="Click to view the MathML source">q∈(1,∞). Furthermore, we prove that the dimension function a4dcf856a4abec86ad496" title="Click to view the MathML source">D:(1,∞)→[0,1] is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in b106ed99a94c7df455bdf20797" title="Click to view the MathML source">(q′,∞), where e5b0ffadd083c7" title="Click to view the MathML source">q′ denotes the Komornik–Loreti constant: although 85a9b0c24eebd7dab1d337bc79" title="Click to view the MathML source">D(q)>D(q′) for all 85b9772bc96a8a35a49" title="Click to view the MathML source">q>q′, we have e6ab4899bac62c111cd20d" title="Click to view the MathML source">D′<0 a.e. in b106ed99a94c7df455bdf20797" title="Click to view the MathML source">(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 85f6" title="Click to view the MathML source">U of bases in which e7c003167b7a6" title="Click to view the MathML source">x=1 has a unique expansion.