ACTA issues

Multiple common expansions in non-integer bases

Vilmos Komornik, Marco Pedicini, Attila Pethő

Acta Sci. Math. (Szeged) 83:1-2(2017), 51-60

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.)