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On the distribution of apk modulo one

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  26 February 2010

R. C. Baker  and
G. Harman
R. C. Baker
Affiliation:
Professor R. C. Baker, Department of Mathematics, Royal Holloway and Bedford New College, Egham, Surrey, TW20 OEX
G. Harman
Affiliation:
Dr. Glyn Harman, School of Mathematics, University of Wales, College of Cardiff, Senghenydd Road, Cardiff, CF2 4AG
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Extract

The problem concerning the distribution of the fractional parts of the sequence ank (k an integer exceeding one) was first considered by Hardy and Littlewood [6] and Weyl [20] earlier this century. This work was developed, with the focus on small fractional parts of the sequence, by Vinogradov [17], Heilbronn [13] and Danicic [2] (see [1]). Recently Heath-Brown [12] has improved the unlocalized versions of these results for k ≥ 6 (a slightly stronger result than Heath-Brown's for K = 8 is given on page 24 of [8]. The method mentioned there can, after some numerical calculation, improve Heath-Brown's result for 8 ≤ k ≤ 20, but still stronger results have recently been obtained by Dr. T. D. Wooley). The cognate problem regarding the sequence apk, where p denotes a prime, has also received some attention. In this situation even the case k = 1 proves to be difficult (see [9] and [14]). The first results in this field were given by Vinogradov (see Chapter 11 of [19] for the case k = 1, [18] for k ≥ 2). For k = 2 the best result to date has been supplied by Ghosh [5], and for ≥, by Harman (Theorem 3 in [9], building on the work in [7] and [8]). In this paper we shall improve the known results for 2 ≤ k ≤ 12. For larger k, Theorem 3 in [8] is more efficient. The theorem we prove is as follows.

MSC classification

Secondary: 11J71: Distribution modulo one
Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 170 - 184
Copyright
Copyright © University College London 1991

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