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 ∈ J q : = [ 0 , ∑ i = 1 ∞ m q − i ] can be written as x = ∑ i = 1 ∞ c i q − i with integer coefficients 0 ⩽ c i < q . If q is a non-integer, then almost every x ∈ 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 ∈ 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 ) .