ar X iv : m at h / 06 09 70 8 v 3 [ m at h . N T ] 1 1 Ju l 2 00 7 UNIQUE EXPANSIONS OF REAL NUMBERS

It was discovered some years ago that there exist non-integer real numbers q > 1 for which only one sequence (ci) of integers 0 ≤ ci < q satisfies the equality P ∞ i=1 ciq −i = 1. The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed q > 1 the set Uq of real numbers x having a unique representation of the form P ∞ i=1 ciq −i = x with integers 0 ≤ ci < q. We carry out a detailed topological study of these sets. In particular, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set U ′ q consisting of all sequences (ci) of integers 0 ≤ ci < q such that P ∞ i=1 ciq −i ∈ Uq . We determine the bases q > 1 for which U ′ q is constant in a neighborhood of q and the bases q > 1 for which U ′ q is a subshift or a subshift of finite type.