Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415479 | Journal of Number Theory | 2015 | 18 Pages |
Let qâ(1,2) and xâ[0,1qâ1]. We say that a sequence (ϵi)i=1ââ{0,1}N is an expansion of x in base q (or a q-expansion) ifx=âi=1âϵiqâi. Let Bâµ0 denote the set of q for which there exists x with exactly âµ0 expansions in base q. In [5] it was shown that minâ¡Bâµ0=1+52. In this paper we show that the smallest element of Bâµ0 strictly greater than 1+52 is qâµ0â1.64541, the appropriate root of x6=x4+x3+2x2+x+1. This leads to a full dichotomy for the number of possible q-expansions for qâ(1+52,qâµ0). We also prove some general results regarding Bâµ0â©[1+52,qf], where qfâ1.75488 is the appropriate root of x3=2x2âx+1. Moreover, the techniques developed in this paper imply that if xâ[0,1qâ1] has uncountably many q-expansions then the set of q-expansions for x has cardinality equal to that of the continuum, this proves that the continuum hypothesis holds when restricted to this specific case.