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THE DISTRIBUTION OF αp MODULO ONE

Published online by Cambridge University Press:  13 February 2002

D.R. HEATH-BROWN
Affiliation:
Mathematical Institute, 24–29 St. Giles', Oxford OX1 3LB [email protected]
C. JIA
Affiliation:
Institute of Mathematics, Academia Sinica, Beijing 100080, P. R. [email protected]
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Abstract

In this paper the authors prove the following result.

Let $\alpha$ be an irrational number. Then for any $\varepsilon > 0$, there are infinitely many prime numbers p such that $\| \alpha p \| < p^{-16/49 +\varepsilon}$.

The exponent $\frac{16}{49}$ improves on $\frac{9}{28}$, which was obtained recently by the second author [Sci. China Ser. A 43 (2000) 703-721]. The result is very close to the exponent $\frac{1}{3}$, which can be obtained under the Generalized Riemann Hypothesis.

Previous approaches to this problem have all used the same basic estimates for the trigonometric sums that arise. However, the present proof uses new bounds, which depend on the Kloosterman sum and also on a counting problem in the geometry of numbers. In addition new techniques for the sieve method are applied. The most significant feature of the new approach is that, unlike previous methods, the exponential sum estimates remain non-trivial for the exponent $\frac{1}{3}$. This gives one hope for an unconditional result as good as that available under the Generalized Riemann Hypothesis.

2000 Mathematical Subject Classification: 11N36, 11J71.

Type
Research Article
Copyright
2002 London Mathematical Society

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