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Powers of Rational Numbers Modulo 1 Lying in Short Intervals

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Abstract

Let p > q > 1 be two coprime integers. We construct some positive numbers ξ such that the numbers ξ(p/q)n, n = 0, 1, 2, . . . , modulo 1 all lie in a short interval. Our results imply, for instance, that there exist three positive real numbers ξ, ζ, τ such that the inequalities ||ξ(5/3)n|| < 2/5, ||ζ (5/3)n|| > 1/10 and ||τ (3/2)2n|| < 14/45 hold for each integer \({n \geqslant 0}\).

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania

    Artūras Dubickas

  2. Institute of Mathematics and Informatics, Akademijos 4, 08663, Vilnius, Lithuania

    Artūras Dubickas

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  1. Artūras Dubickas
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Correspondence to Artūras Dubickas.

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Dubickas, A. Powers of Rational Numbers Modulo 1 Lying in Short Intervals. Results. Math. 57, 23–31 (2010). https://doi.org/10.1007/s00025-009-0001-0

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  • DOI: https://doi.org/10.1007/s00025-009-0001-0

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