Expansions in multiple bases
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EXPANSIONS IN MULTIPLE BASES Y.-Q. LI Institut de Math´ ematiques de Jussieu – Paris Rive Gauche, Sorbonne Universit´ e – Campus Pierre et Marie Curie, Paris, 75005, France e-mails: [email protected], [email protected] School of Mathematics, South China University of Technology, Guangzhou, 510641, P.R. China e-mails: [email protected], [email protected] (Received April 18, 2020; revised May 16, 2020; accepted July 20, 2020)
Abstract. Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy, lazy, quasi-lazy and unique expansions in multiple bases.
1. Introduction As is well known, expansion in a given base is the most common way to represent a real number. For example, expansions in base 10 are used in our daily lives and expansions in base 2 are used in computer systems. Expansions of real numbers in integer bases have been widely used. As a natural generalization, R´enyi [32] introduced expansions in non-integer bases, which attracted a lot of attention in the following decades. Until Neunh¨auserer [29] began the study of expansions in two bases recently, all expansions studied were in one base. In this paper, we begin the study of expansions in multiple bases. Let N be the set of positive integers {1, 2, 3, . . .} and R be the set of real numbers. We recall the concept of expansions in one base first. Let m ∈ N, β ∈ (1, m + 1] and x ∈ R. A sequence w = (wi )i≥1 ∈ {0, 1, . . . , m}N is called a β-expansion of x if x = πβ (w) :=
∞ wi i=1
βi
.
The author is grateful to the Oversea Study Program of Guangzhou Elite Project (GEP) for financial support (JY201815). Key words and phrases: expansion, multiple bases, greedy, lazy, quasi-greedy, quasi-lazy. Mathematics Subject Classification: primary 11A63, secondary 37B10.
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
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Y.-Q. LI
m It is known that x has a β-expansion if and only if x ∈ [0, β−1 ] (see for example [3–5,32]). The following question is natural to be thought of: Given m ∈ N, β0 , β1 , . . . , βm > 1, x ∈ R and (wi )i≥1 ∈ {0, 1, . . . , m}N , in which case should we say that (wi )i≥1 is a (β0 , β1 , . . . , βm )-expansion of x, such that when β0 , wi β1 , . . . , βm are taken to be the same β, we have x = ∞ i=1 β i ? Proposition 1.1 may answer this question. Let us give some notations first. For all m ∈ N and β0 , β1 , . . . , βm > 1, we define
ak :=
k βk
and bk :=
Note that a0 = 0 and bm =
k m + βk βk (βm − 1) m βm −1 .
for all k ∈ {0, . . . , m}.
For all m ∈ N, let
Dm := (β0 , . . . , βm ) : β0 , . . . , βm > 1 and
ak < ak+1 ≤ bk < bk+1 for all k, 0 ≤ k ≤ m − 1 .
It is worth to note that Dm is large enough to ensure that (β, . . . , β ) ∈ Dm
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