On the number of unique expansions in non-integer bases

Let q > 1 be a real number and let m = m(q) be the largest integer smaller than q. It is well known that each number x is an element of Jq := [0. Sigma(infinity)(i=1) mq(-i)] can be written as x = Sigma(infinity)(i=1)c(i)q(-i) with integer coefficients 0 <= c(i) < q. If q is a non-integer, then almost every x is an element of J(q) has continuum many expansions of this form. In this note we consider some properties of the set U(q) consisting of numbers x is an element of J(q) having a unique representation of this form. More specifically, we compare the size of the sets U(q) and U(r) for values q and r satisfying 1 < q < r and m(q) = m(r). (C) 2008 Elsevier B.V. All rights reserved.