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 qiϵ I each x ϵ J has common expansions for some bases q1 < ⋯ qr.

Komornik, V., Pedicini, M., Petho, A.T. (2017). Multiple common expansions in non-integer bases. ACTA SCIENTIARUM MATHEMATICARUM, 83(1-2), 51-60 [10.14232/actasm-015-080-0].

Multiple common expansions in non-integer bases

Pedicini, Marco;Petho, Attila Tamas
2017

Abstract

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 qiϵ I each x ϵ J has common expansions for some bases q1 < ⋯ qr.
Komornik, V., Pedicini, M., Petho, A.T. (2017). Multiple common expansions in non-integer bases. ACTA SCIENTIARUM MATHEMATICARUM, 83(1-2), 51-60 [10.14232/actasm-015-080-0].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/327022
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