On the topological structure of univoque sets

Abstract

Erdős, Horváth and Joó discovered some years ago that for some real numbers 1 < q < 2 there exists only one sequence c i of zeroes and ones such that ∑ c i q − i = 1 . Subsequently, the set U of these numbers was characterized algebraically in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions 1 = ∑ q − n i and related problems, Bull. Soc. Math. France 118 (1990) 377–390] and [V. Komornik, P. Loreti, Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar. 44 (2) (2002) 195–216]. We establish an analogous characterization of the closure U ¯ of U . This allows us to clarify the topological structure of these sets: U ¯ ∖ U is a countable dense set of U ¯ , so the latter set is perfect. Moreover, since U is known to have zero Lebesgue measure, U ¯ is a Cantor set.