### Abstract

We investigate the existence of simultaneous representations of real numbers x in bases 1 < q_{1} < ⋯ < q_{r}, r ≥ 2, with a finite digit set A ⊃ ℝ. We prove that if A contains both positive and negative digits, then each real number has infinitely many common expansions. In general the bases depend on x. If A contains the digits -1,0,1, then there exist two non-empty open intervals I, J such that for any fixed q_{i} ϵ I each x ϵ J has common expansions for some bases q1 < ⋯ q_{r}.

Original language | English |
---|---|

Pages (from-to) | 51-60 |

Number of pages | 10 |

Journal | Acta Scientiarum Mathematicarum |

Volume | 83 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Interval filling sequences
- Simultaneous Rényi expansions

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Acta Scientiarum Mathematicarum*,

*83*(1-2), 51-60. https://doi.org/10.14232/actasm-015-080-0

**Multiple common expansions in non-integer bases.** / Komornik, Vilmos; Pedicini, Marco; Pethő, A.

Research output: Contribution to journal › Article

*Acta Scientiarum Mathematicarum*, vol. 83, no. 1-2, pp. 51-60. https://doi.org/10.14232/actasm-015-080-0

}

TY - JOUR

T1 - Multiple common expansions in non-integer bases

AU - Komornik, Vilmos

AU - Pedicini, Marco

AU - Pethő, A.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We investigate the existence of simultaneous representations of real numbers x in bases 1 < q1 < ⋯ < qr, r ≥ 2, with a finite digit set A ⊃ ℝ. We prove that if A contains both positive and negative digits, then each real number has infinitely many common expansions. In general the bases depend on x. If A contains the digits -1,0,1, then there exist two non-empty open intervals I, J such that for any fixed qi ϵ I each x ϵ J has common expansions for some bases q1 < ⋯ qr.

AB - We investigate the existence of simultaneous representations of real numbers x in bases 1 < q1 < ⋯ < qr, r ≥ 2, with a finite digit set A ⊃ ℝ. We prove that if A contains both positive and negative digits, then each real number has infinitely many common expansions. In general the bases depend on x. If A contains the digits -1,0,1, then there exist two non-empty open intervals I, J such that for any fixed qi ϵ I each x ϵ J has common expansions for some bases q1 < ⋯ qr.

KW - Interval filling sequences

KW - Simultaneous Rényi expansions

UR - http://www.scopus.com/inward/record.url?scp=85020995170&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020995170&partnerID=8YFLogxK

U2 - 10.14232/actasm-015-080-0

DO - 10.14232/actasm-015-080-0

M3 - Article

AN - SCOPUS:85020995170

VL - 83

SP - 51

EP - 60

JO - Acta Scientiarum Mathematicarum

JF - Acta Scientiarum Mathematicarum

SN - 0001-6969

IS - 1-2

ER -