On univoque and strongly univoque sets

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• 作者：Pieter C. Allaart ^{allaart@unt.edu}
• 关键词：primary ; 11A63 ; secondary ; 28A78
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：21 February 2017
• 年：2017
• 卷：308
• 期：Complete
• 页码：575-598
• 全文大小：469 K
• 卷排序：308

文摘

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet {0,1,…,α}{0,1,…,α} and a real number (base) 1<β<α+11<β<α+1, the so-called univoque set of numbers which have a unique expansion in base β has garnered a great deal of attention in recent years. Motivated by recent applications of β-expansions to Bernoulli convolutions and a certain class of self-affine functions, we introduce the notion of a strongly univoque   set. We study in detail the set DβDβ of numbers which are univoque but not strongly univoque. Our main result is that DβDβ is nonempty if and only if the number 1 has a unique nonterminating expansion in base β  , and in that case, DβDβ is uncountable. We give a sufficient condition for DβDβ to have positive Hausdorff dimension, and show that, on the other hand, there are infinitely many values of β   for which DβDβ is uncountable but of Hausdorff dimension zero.