Finite beta-expansions

Frougny, Christiane; Solomyak, Boris

doi:10.1017/s0143385700007057

Cited by 191 publications

(145 citation statements)

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“…closed unit disc |z| 1 with at least one conjugate lying on the circle |z| = 1). Finite β-expansions have been studied in [9] and [11]; those results are also related to Pisot numbers.…”

Section: Introductionmentioning

confidence: 99%

On Integer Sequences Generated by Linear Maps

Dubickas¹

2009

Glasgow Math. J.

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Abstract. Let x 0 < x 1 < x 2 < · · · be an increasing sequence of positive integers given by the formula x n = βx n−1 + γ for n = 1, 2, 3, . . . , where β > 1 and γ are real numbers and x 0 is a positive integer. We describe the conditions on integers b d , . . . , b 0 , not all zero, and on a real number β > 1 under which the sequence of integers w n = b d x n+d + · · · + b 0 x n , n = 0, 1, 2, . . . , is bounded by a constant independent of n. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequence qx n+1

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“…closed unit disc |z| 1 with at least one conjugate lying on the circle |z| = 1). Finite β-expansions have been studied in [9] and [11]; those results are also related to Pisot numbers.…”

Section: Introductionmentioning

confidence: 99%

On Integer Sequences Generated by Linear Maps

Dubickas¹

2009

Glasgow Math. J.

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“…Frougny and Solomyak [18] have studied situations in which one can guarantee that a given Pisot number is a simple beta-number.…”

Section: Introductionmentioning

confidence: 99%

On beta expansions for Pisot numbers

Boyd

1996

Math. Comp.

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Abstract. Given a number β > 1, the beta-transformation T = T β is defined for x ∈ [0, 1] by T x := βx (mod 1). The number β is said to be a betanumber if the orbit {T n (1)} is finite, hence eventually periodic. In this case β is the root of a monic polynomial R(x) with integer coefficients called the characteristic polynomial of β. If P(x) is the minimal polynomial of β, then R(x) = P(x)Q(x) for some polynomial Q(x). It is the factor Q(x) which concerns us here in case β is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether Q(x) must be cyclotomic in this case, particularly if 1 < β < 2. We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in [1, 1.9324] ∪ [1.9333, 1.96] (an infinite set), by a search up to degree 50 in [1.9, 2], to degree 60 in [1.96, 2], and to degree 20 in [2, 2.2]. We find the smallest counterexample, the counterexample of smallest degree, examples where Q(x) is nonreciprocal, and examples where Q(x) is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to 2 from above, and infinite sequences of β with Q(x) nonreciprocal which converge to 2 from below and to the 6th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in [1,2]. The Pisot numbers for which Q(x) is cyclotomic are related to an interesting closed set of numbers F introduced by Flatto, Lagarias and Poonen in connection with the zeta function of T . Our examples show that the set S of Pisot numbers is not a subset of F.

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“…However, there exist Pisot numbers that do not fullfill (F), such as all numbers with non purely periodic quasi-greedy β-expansion of 1. In [7], it is also shown that if…”

Section: Introductionmentioning

confidence: 99%

“…In [10], and before in [8], the authors posed the question whether Hypothesis B, introduced in [8], and the finiteness property (F), introduced in [7], are equivalent. Hypothesis B is a condition on the carries of the digits in the expansion of positive integers in a base system defined by a linear recurrence.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Linear Recursive Odometers and Beta-Expansions

Iacò

Steiner

Tichy

2016

Uniform Distribution Theory

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ABSTRACT. The aim of this paper is to study the connection between different properties related to β-expansions. In particular, the relation between two conditions, both ensuring purely discrete spectrum of the odometer, is analyzed. The first one is the so-called Hypothesis B for the G-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Pisot β-numerations.

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Finite beta-expansions

Abstract: We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.

Cited by 191 publications

References 12 publications

On Integer Sequences Generated by Linear Maps

On Integer Sequences Generated by Linear Maps

On beta expansions for Pisot numbers

Linear Recursive Odometers and Beta-Expansions

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