Hausdorff dimension of univoque sets and Devil's staircase

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• 作者：Vilmos Komornik^a ; ^{komornik@math.unistra.fr" class="auth_mail" title="E-mail the corresponding author} ; Derong Kong^b ; ^{derongkong@126.com" class="auth_mail" title="E-mail the corresponding author} ; Wenxia Li^c ; ^{wxli@math.ecnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11A63 ; secondary ; 10K50 ; 11K55 ; 37B10 ; 28A80
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：165-196
• 全文大小：500 K

文摘

We fix a positive integer M  , and we consider expansions in arbitrary real bases 816312804&_mathId=si1.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=d742e3d8cc3908ad5d3181696fd55d23" title="Click to view the MathML source">q>1$q>1$ over the alphabet 816312804&_mathId=si15.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=6730daadf0ebc97b1668e7ed06dd978d" title="Click to view the MathML source">{0,1,…,M}$\{0,1,\dots ,M\}$. We denote by 816312804&_mathId=si107.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=4b193b0c769459ba20dee874ea4da3d9" title="Click to view the MathML source">U_q${\mathcal{U}}_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension 816312804&_mathId=si4.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=066a133e3784126e947feaaa5083a6d1" title="Click to view the MathML source">D(q)$D(q)$ of 816312804&_mathId=si107.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=4b193b0c769459ba20dee874ea4da3d9" title="Click to view the MathML source">U_q${\mathcal{U}}_q$ for each 816312804&_mathId=si6.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=acbf822e516c827a07fce8d9bf05dfe8" title="Click to view the MathML source">q∈(1,∞)$q\in (1,\infty )$. Furthermore, we prove that the dimension function 816312804&_mathId=si7.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=c23c431e1eda4dcf856a4abec86ad496" title="Click to view the MathML source">D:(1,∞)→[0,1]$D:(1,\infty )\to [0,1]$ is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in 816312804&_mathId=si25.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=7d8d76b106ed99a94c7df455bdf20797" title="Click to view the MathML source">(q^′,∞)$(q^{\prime },\infty )$, where 816312804&_mathId=si9.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=836fe880a27b6ee451e5b0ffadd083c7" title="Click to view the MathML source">q^′$q^{\prime }$ denotes the Komornik–Loreti constant: although 816312804&_mathId=si10.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=eca58485a9b0c24eebd7dab1d337bc79" title="Click to view the MathML source">D(q)>D(q^′)$D(q)>D(q^{\prime })$ for all 816312804&_mathId=si11.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=73d339bf1d3bc85b9772bc96a8a35a49" title="Click to view the MathML source">q>q^′$q>q^{\prime }$, we have 816312804&_mathId=si12.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=698496d0d9e6ab4899bac62c111cd20d" title="Click to view the MathML source">D^′<0$D^{\prime }<0$ a.e. in 816312804&_mathId=si25.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=7d8d76b106ed99a94c7df455bdf20797" title="Click to view the MathML source">(q^′,∞)$(q^{\prime },\infty )$. 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 816312804&_mathId=si13.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=3a8e11268bcd9f69004f6f6c2bbb85f6" title="Click to view the MathML source">U$\mathcal{U}$ of bases in which 816312804&_mathId=si14.gif&_user=111111111&_pii=S0001870816312804&_rdoc=1&_issn=00018708&md5=b41753bc14e1649565ae7c003167b7a6" title="Click to view the MathML source">x=1$x=1$ has a unique expansion.