Digit frequencies of beta-expansions
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DIGIT FREQUENCIES OF BETA-EXPANSIONS Y.-Q. LI Institut de Math´ ematiques de Jussieu – Paris Rive Gauche, Sorbonne Universit´ e, Campus Pierre et Marie Curie, Paris, 75005, France School of Mathematics, South China University of Technology, Guangzhou, 510641, P. R. China e-mails: [email protected], [email protected], [email protected] (Received November 14, 2019; revised January 18, 2020; accepted January 20, 2020)
Abstract. Let β > 1 be a non-integer. First we show that Lebesgue almost every number has a β-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many β-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced β-expansions, where an infinite sequence on the finite alphabet {0, 1, . . . , m} is called balanced if the frequency of the digit k is equal to the frequency of the digit m − k for all k ∈ {0, 1, . . . , m}. Finally we consider variable frequency and prove that for every pseudo-golden ratio β ∈ (1, 2), there exists a constant c = c(β) > 0 such that for any p ∈ [ 12 − c, 21 + c], Lebesgue almost every x has infinitely many β-expansions with frequency of zeros equal to p.
1. Introduction To represent real numbers, the most common way is to use expansions in integer bases, especially in base 2 or 10. As a natural generalization, expansions in non-integer bases were introduced by R´enyi [26], and then attracted a lot of attention until now (see for example [1,2,8,11,17,18,23–25, 27,28]). They are known as beta-expansions nowadays. Let N = {1, 2, 3, . . .} be the set of positive integers and R be the set of real numbers. For β > 1, we define the alphabet by Aβ = 0, 1, . . . , ⌈β⌉ − 1 .
where ⌈β⌉ denotes the smallest integer no less than β, and similarly we use ⌊β⌋ to denote the greatest integer no larger than β. Let x ∈ R. A sequence (εi )i≥1 ∈ AN β is called a β-expansion of x if x=
∞ εi . βi i=1
Key words and phrases: β-expansion, digit frequency, pseudo-golden ratio. Mathematics Subject Classification: primary 11A63, secondary 11K55. c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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Y.-Q. Y.-Q. LI LI
o For β > 1, let Iβ be the interval [0, ⌈β⌉−1 β−1 ], and let Iβ be the interior of Iβ (i.e.
Iβo = (0, ⌈β⌉−1 β−1 )). It is straightforward to check that x has a β-expansion if and only if x ∈ Iβ . An interesting phenomenon is that an x may have many √ β-expansions. For example, [14, Theorem 3] shows that if β ∈ (1, 1+2 5 ), every x ∈ Iβo has a continuum of different β-expansions, and [29, Theorem 1] shows that if β ∈ (1, 2), Lebesgue almost every x ∈ Iβ has a continuum of different β-expansions. For more on the cardinality of β-expansions, we refer the reader to [7,15,19]. In this paper we focus on the digit frequencies of β-expansions, which is a classical research topic. For example, Borel’s normal number theorem [9] says that for any integer β > 1, Lebesgue almost every x ∈ [0, 1] has a β-expan
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