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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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References

1 Baker, R. C.. Diophantine Inequalities, London Math. Soc. Monographs N.S. 1 (Oxford Science Publications, 1986).Google Scholar
2 Danicic, I.. Contributions to Number Theory, Ph.D. Thesis (London, 1957).Google Scholar
3 Davenport, H.. Multiplicative Number Theory, Second edition, revised by H. L. Montgomery (Springer, New York-Heidelberg-Berlin, 1980).CrossRefGoogle Scholar
4 Erdös, P. and Szekeres, E.. Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwendtes zahlentheoretisches Problem. Acta Sci. Math. Szeged., 7 (1934), 95–102.Google Scholar
5 Ghosh, A.. The distribution of ap2 modulo one. Proc. London Math. Soc. (3), 42 (1981), 252–269.CrossRefGoogle Scholar
6 Hardy, G. H. and Littlewood, J. E.. Some problems of Diophantine approximation I. On the fractional part of nkλ. Acta Math, 37 (1914), 155–191.CrossRefGoogle Scholar
7 Harman, G.. Trigonometric sums over primes I. Mathematika, 28 (1981), 249–254.CrossRefGoogle Scholar
8 Harman, G.. Trigonometric sums over primes II. Glasgow Math. J., 24 (1983), 23–37.CrossRefGoogle Scholar
9 Harman, G.. On the distribution of ap modulo one. J. London Math. Soc. (2), 27 (1983), 9–18.CrossRefGoogle Scholar
10 Heath-Brown, D. R.. Diophantine approximation with square-free integers. Math Zeit., 187 (1984), 335–344.CrossRefGoogle Scholar
11 Heath-Brown, D. R.. Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math, 34 (1982), 1365–1377.CrossRefGoogle Scholar
12 Heath-Brown, D. R.. The fractional part of ank. Mathematika, 35 (1988), 28–37.CrossRefGoogle Scholar
13 Heilbronn, H.. On the distribution of the sequence n2λ(mod 1). Quart. J. Math. Oxford, 19 (1948), 249–256.CrossRefGoogle Scholar
14 Vaughan, R. C.. On the distribution of ap modulo one. Mathematika, 24 (1977), 135–141.CrossRefGoogle Scholar
15 Vaughan, R. C.. The Hardy-Littlewood Method (Cambridge University Press, 1981).Google Scholar
16 Vaughan, R. C.. Some remarks on Weyl sums. Coloqu. Math. Soc. Janos Bolyai, 34, 1585-1601 (Elsevier, North Holland, Amsterdam, 1984).Google Scholar
17 Vinogradov, I. M.. Analytischer Beweis des Satzes über die Verteilung der Bruchteile eines ganzen Polynoms. Bull. Acad. Sci. USSR (6), 21 (1927), 567–578.Google Scholar
18 Vinogradov, I. M.. A general distribution law for the fractional parts of values of a polynomial with variable running over primes. C.R. (Doklady) Acad.-Sci. URSS (NS), 51 (1946), 491–492.Google Scholar
19 Vinogradov, I. M.. The Method of Trigonometric Sums in the Theory of Numbers. Translated, revised and annotated by Davenport, A. and Roth, K. F. (Interscience, New York, 1954).Google Scholar
20 Weyl, H.. Uber die Gleichverteilung von Zahlen mod Eins. Math. Ann., 11 (1916), 313–352.CrossRefGoogle Scholar