Abstract. We investigate the existence of simultaneous representations of real numbers $x$ in bases $1< q_1<\cdots < q_r$, $r\geq2$, with a finite digit set $A\subset{\msbm R}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 $q_1\in I$ each $x\in J$ has common expansions for some bases $q_1<\cdots < q_r$.
DOI: 10.14232/actasm-015-080-0
AMS Subject Classification
(1991): 11A63, 11B83
Keyword(s):
simultaneous Rényi expansions,
interval filling sequences
Received November 11, 2015, and in revised form February 1, 2016. (Registered under 80/2015.)
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