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On the Distribution Modulo 1 of the Sequence αn2 + βn

Published online by Cambridge University Press:  20 November 2018

Wolfgang M. Schmidt
Wolfgang M. Schmidt*
Affiliation:
University of Colorado, Boulder, Colorado
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Dirichlet's Theorem says that for any real α and for N ≧ 1, there exists a natural n ≦ N with

where || || denotes the distance to the nearest integer. Heilbronn [2], improving estimates of Vinogradov [3], showed that for α, N as above and for ϵ ≧ 0, there exists an n ≦ N with

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 4 , 01 August 1977 , pp. 819 - 826
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Davenport, H., On a theorem of Heilbronn, Quart. J. Math. (2) 18 (1967), 337–344.Google Scholar
2. Heilbronn, H., On the distribution of the sequence n∼d (mod 1), Quart. J. Math. 10 (1948), 249–2.56.Google Scholar
3. Vinogradov, I. M., Analytischer Beweis des Satzes iiber die Verteilung der Bruchteile eines ganzen Polynonis, Bull. Acad. Sci. USSR (6) 21 (1927), 567–578.Google Scholar
4. Vinogradov, I. M., The method of trigonometric sums in the theory of numbers (1947), English transi. (Interscience, New York, 1954).Google Scholar