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.

On the topological structure of univoque sets / Vilmos, Komornik; Loreti, Paola. - In: JOURNAL OF NUMBER THEORY. - ISSN 0022-314X. - STAMPA. - 122:1(2007), pp. 157-183. [10.1016/j.jnt.2006.04.006]

On the topological structure of univoque sets

LORETI, Paola
2007

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.
2007
beta-expansion; cantor sets; expansions in non-integer bases; golden number; non-integer bases; thue-morse sequence
01 Pubblicazione su rivista::01a Articolo in rivista
On the topological structure of univoque sets / Vilmos, Komornik; Loreti, Paola. - In: JOURNAL OF NUMBER THEORY. - ISSN 0022-314X. - STAMPA. - 122:1(2007), pp. 157-183. [10.1016/j.jnt.2006.04.006]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/14218
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