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.
- Interval filling sequences
- Simultaneous Rényi expansions
ASJC Scopus subject areas
- Applied Mathematics