کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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6425046 | 1633783 | 2017 | 32 صفحه PDF | دانلود رایگان |
We fix a positive integer M, and we consider expansions in arbitrary real bases q>1 over the alphabet {0,1,â¦,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 D(q) of Uq for each qâ(1,â). Furthermore, we prove that the dimension function D:(1,â)â[0,1] is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in (qâ²,â), where qâ² denotes the Komornik-Loreti constant: although D(q)>D(qâ²) for all q>qâ², we have Dâ²<0 a.e. in (qâ²,â). During the proofs we improve and generalize a theorem of ErdÅs 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.
Journal: Advances in Mathematics - Volume 305, 10 January 2017, Pages 165-196