Powers of Rational Numbers Modulo 1 Lying in Short Intervals

Dubickas, Artūras

doi:10.1007/s00025-009-0001-0

Cited by 4 publications

(5 citation statements)

References 11 publications

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“…For an overview see [62]; recent contributions are e.g. due to Akiyama, Frougny and Sakarovitch [8,9], Dubickas [28,29] and Kaneko [42,43].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Quantitative uniform distribution results for geometric progressions

Aistleitner

2014

Isr. J. Math.

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By a classical theorem of Koksma the sequence of fractional parts ({x n }) n≥1 is uniformly distributed for almost all values of x > 1. In the present paper we obtain an exact quantitative version of Koksma's theorem, by calculating the precise asymptotic order of the discrepancy of ({ξx sn }) n≥1 for typical values of x (in the sense of Lebesgue measure). Here ξ > 0 is an arbitrary constant, and (s n ) n≥1 can be any increasing sequence of positive integers.

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“…For an overview see [62]; recent contributions are e.g. due to Akiyama, Frougny and Sakarovitch [8,9], Dubickas [28,29] and Kaneko [42,43].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Quantitative uniform distribution results for geometric progressions

Aistleitner

2014

Isr. J. Math.

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“…has a solution α = 0 for any fixed odd p ≥ 3. As remarked in [11], it follows from ( 16) that the bound in ( 18) cannot be improved to p −1 − 4p −3 < τ (p/1). Finally, the bound from Theorem 3.12 can be slightly improved for q odd with a similar method.…”

Section: 5ºmentioning

confidence: 89%

“…It would be nice to have cardinality equal to |R| instead of greater |Z| on the right hand sides in (11), (12). If we assume the continuum hypothesis to be true (which is known to be undecidable due to P. Cohen), then indeed we may make this replacement.…”

Section: 2ºmentioning

confidence: 99%

Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

Schleischitz

2017

Uniform Distribution Theory

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ABSTRACT. This paper deals with the distribution of αζ n mod 1, where α = 0, ζ > 1 are fixed real numbers and n runs through the positive integers. Denote by . the distance to the nearest integer. We investigate the case of αζ n all lying in prescribed small intervals modulo 1 for all large n, with focus on the case αζ n ≤ for small > 0. We are particularly interested in what we call cardinality gap phenomena. For example for fixed ζ > 1 and small > 0 there are at most countably many values of α such that αζ n ≤ for all large n, whereas larger induces an uncountable set. We investigate the value of at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational ζ > 1. Results concerning Pisot and Salem numbers such as some contribution to Mahler's 3/2-problem are implicitly deduced. We study similar questions for fixed α = 0 as well. Communicated by Michael Drmota

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“…Furthermore, Dubickas also proved that for coprime p > q > 1, there exists ξ > 0 such that {ξ(p/q) n } lies in a short interval of [0, 1]. In particular, his work implies that {ξ(3/2) 2n } < 14/45 [5].…”

Section: Related Workmentioning

confidence: 96%

On the Uniformity of $(3/2)^n$ Modulo 1

Neeley,

Taylor-Rodriguez,

Veerman

et al. 2018

Preprint

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It has been conjectured that the sequence (3/2) n modulo 1 is uniformly distributed. The distribution of this sequence is significant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we describe an algorithm to compute (3/2) n modulo 1 to n = 10 8 . We then statistically analyze its distribution. Our results strongly agree with the hypothesis that (3/2) n modulo 1 is uniformly distributed.

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Powers of Rational Numbers Modulo 1 Lying in Short Intervals

Cited by 4 publications

References 11 publications

Quantitative uniform distribution results for geometric progressions

Quantitative uniform distribution results for geometric progressions

Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

On the Uniformity of $(3/2)^n$ Modulo 1

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