Developments in non-integer bases

P. Erdös, V. Komornik

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Abstract

We prove various theorems concerning the developments in non-integer bases. We mention two of them here, which answer some questions formulated several years ago. First fix a real number q > 1 and consider the increasing sequence 0 = y 0 < y 1 < y 2 < ··· of those real numbers y which have at least one representation of the form y = ε 0 + ε 1 q + ··· + ε n q n with some integer n ≧ 0 and coefficients ε i ∈ {0,1}. Then the difference sequence y k+1 - y k tends to 0 for all q, sufficiently close to 1. Secondly, for each q, sufficiently close to 1, there exists a sequence (ε i ) of zeroes and ones, satisfying ∑∞ i=1 ε i q -i =1 as formula presented and containing all possible finite variations of the digits 0 and 1.

Original languageEnglish
Pages (from-to)57-83
Number of pages27
JournalActa Mathematica Hungarica
Volume79
Issue number1-2
Publication statusPublished - Apr 1998

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ASJC Scopus subject areas

  • Mathematics(all)

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