On the topological structure of univoque sets

Abstract

Erdos, Horvath and Joo 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 Sigma c(i) q(-i) = 1. Subsequently, the set U of these numbers was characterized algebraically in [P. Erdos, I. Joo, V. Komornik, Characterization of the unique expansions 1 = Sigma q(-ni) 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) over bar of U. This allows us to clarify the topological structure of these sets: (U) over bar U is a countable dense set of (U) over bar, so the latter set is perfect. Moreover, since U is known to have zero Lebesgue measure, (U) over bar is a Cantor set. (C) 2006 Elsevier Inc. All rights reserved

Generalized golden ratios of ternary alphabets

“…Much research was stimulated by the discovery of Erdős, Horváth and Joó [5] who proved the existence of many real numbers 1 < q < 2 for which only one sequence (c i ) of zeroes and ones satisfies the equality The set of such "univoque" bases has a fractal nature; see, e.g., [6], [8], [10], where arbitrary bases q > 1 are also considered.…”

A property of algebraic univoque numbers

“…Our proof of Theorem 1.1 relies on a characterization of the closure U of U, recently obtained by Komornik and Loreti in [9] In the sequel, a sequence always means a sequence of nonnegative integers. We use systematically the lexicographical order between sequences; we write (en) < (h) if there exists an index n ^ 1 such that a* = bi for i<n and a n <b n .…”

“…The following algebraic characterization of the set U can be found in [5], [6], [9]: THEOREM 2.1. The map <?>->• (7; (<?))…”

“…Subsequently, the set U of such univoque numbers was characterized in [5], [6], [9] (see Theorem 2.1). Using this characterization, Komornik and Loreti showed in [7] that U has a smallest element q' ^1.787 and the corresponding expansion (r,) is given by the truncated Thue-Morse sequence, defined by setting T 2 N =1 for N =0,1,... and…”

Univoque bases and Hausdorff dimension

“…Recently, Komornik and Loreti showed in [18] that its closure U is a Cantor set (see also, [9]), i.e., a nonempty closed set having neither isolated nor interior points. Writing the open set (1, M + 1]\U = (1, M + 1)\U as the disjoint union of its connected components, i.e.,…”