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On the fractional parts of αn2 and βn

Published online by Cambridge University Press:  18 May 2009

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham, Surrey
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We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such that

This extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 2 , July 1981 , pp. 181 - 183
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

REFERENCES

1.Baker, R. C., Recent results on fractional parts of polynomials, Number theory, Carbondale 1979, Lecture Notes in Mathematics No. 751 (Springer-Verlag, 1979), 10–18.CrossRefGoogle Scholar
2.Baker, R. C. and Gajraj, J., Some non-linear Diophantine approximations. Acta Arith. 31 (1976), 325–341.CrossRefGoogle Scholar
3.Davenport, H., On a theorem of Heilbronn, Quart. J. Math. Oxford Ser. 2, 18 (1967), 339–344.CrossRefGoogle Scholar
4.Heilbronn, H., On the distribution of the sequence n2θ(mod 1), Quart. J. Math. Oxford Ser. 1, 19 (1948), 249–256.CrossRefGoogle Scholar
5.Schmidt, W. M., On the distribution modulo 1 of die sequence αn2 + βn, Canad. J. Math. 29 (1977), 819–826.CrossRefGoogle Scholar