Multiple expansions of real numbers with digits set { 0 , 1 , q}

Abstract

For q> 1 we consider expansions in base q with digits set { 0 , 1 , q}. Let U q be the set of points which have a unique q-expansion. For k= 2 , 3 , … , ℵ let B k be the set of bases q> 1 for which there exists x having precisely k different q-expansions, and for q∈ B k let Uq(k) be the set of all such x’s which have exactly k different q-expansions. In this paper we show that Bℵ0=[2,∞)andBk=(qc,∞)for anyk≥2,where q c ≈ 2.32472 is the appropriate root of x 3 - 3 x 2 + 2 x- 1 = 0. Moreover, we show that for any integer k≥ 2 and any q∈ B k the Hausdorff dimensions of Uq(k) and U q are the same, i.e., dimHUq(k)=dimHUqfor anyk≥2.Finally, we conclude that the set of points having a continuum of q-expansions has full Hausdorff dimension.