Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T06:13:15.307Z Has data issue: false hasContentIssue false
  • English
  • Français

The fractional part of αnk

Published online by Cambridge University Press:  26 February 2010

D. R. Heath-Brown
D. R. Heath-Brown
Affiliation:
Magdalen College, Oxford 0X1 4AU
Get access

Extract

Let α be a real number and k a positive integer. We shall be interested in integer values of n for which ║αnk║ is small. For the case k = 1 we have Dirichlet's Theorem. For any N ≥ 1 there exists n ≤ N with

Type
Research Article
Information
Mathematika , Volume 35 , Issue 1 , June 1988 , pp. 28 - 37
Copyright
Copyright © University College London 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.Heath-Brown, D. R.. Weyl's inequality, Hua's inequality, and Waring's problem. To appear in the J. London Math. Soc.Google Scholar
4.Heilbronn, H.. On the distribution of the sequence n2ν (mod 1). Quart. J. Math. Oxford Ser., 19 (1948), 249–256.CrossRefGoogle Scholar
5.Karatsuba, A. A.. On the function G(n) in Waring's problem. Izv. Akad. Nauk SSSR, Ser. Mat., 49 (1985), 935–947.Google Scholar
6.Roth, K. F.. Rational approximations to algebraic numbers. Mathematika, 2 (1955), 1–20.Google Scholar
7.Vinogradov, I. M.. The Method of Trigonometrical Sums in the Theory of Numbers, translated from the Russian, revised and annotated by Roth, K. F. and Davenport, A. (Interscience, London, 1954).Google Scholar
8.Vinogradov, I. M.. A new estimate for ζ(l + it). Izv. Akad. Nauk SSSR, Ser. Mat., 22 (1958), 161–164.Google Scholar