### 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 language | English |
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Pages (from-to) | 57-83 |

Number of pages | 27 |

Journal | Acta Mathematica Hungarica |

Volume | 79 |

Issue number | 1-2 |

Publication status | Published - Apr 1998 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

*79*(1-2), 57-83.