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Note on a Diophantine inequality in several variables

Author(s): Jeffrey T. Barton; Hugh L. Montgomery; Jeffrey D. Vaaler
Journal: Proc. Amer. Math. Soc. 129 (2001), 337-345.
MSC (2000): Primary 11J25, 11K60, 11K38
Posted: August 28, 2000
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Abstract:

We establish estimates for the number of points that belong to an aligned box in $(\mathbb{R}/\mathbb{Z})^N$ in terms of certain exponential sums. These generalize previous results that were known only in case $N=1$.

References:

1.
Baker, R.C., Diophantine Inequalities, London Mathematical Society Monographs (New Series), Vol. 1, Clarendon Press, Oxford, 1986. MR 88f:11021

2.
Cochran, T., Trigonometric approximation and uniform distribution modulo one, Proc. American Math. Soc. 103 (1988), 695-702. MR 89j:11071

3.
Montgomery, H.L., Ten Lectures on the Interface of Analytic Number Theory and Harmonic Analysis, American Mathematical Society, Providence, RI, 1994. MR 96i:11002

4.
Vaaler, J.D., Some extremal functions in Fourier analysis, Bull. American Math. Soc. 12 (1985), 183-216. MR 86g:42005

5.
Vinogradov, I.M., The Method of Trigonometrical Sums in the Theory of Numbers, translated by H. Davenport & K. F. Roth, Interscience, London, 1954.MR 15:941b

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Additional Information:

Jeffrey T. Barton
Affiliation: Department of Mathematics, Birmingham-Southern College, Birmingham, Alabama 35254
Email: jbarton@bsc.edu

Hugh L. Montgomery
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: hlm@math.lsa.umich.edu

Jeffrey D. Vaaler
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: vaaler@math.utexas.edu

DOI: 10.1090/S0002-9939-00-05795-6
PII: S 0002-9939(00)05795-6
Received by editor(s): April 15, 1999
Posted: August 28, 2000
Additional Notes: The first and third authors' research was supported in part by the National Science Foundation (DMS-9622556) and the Texas Advanced Research Project.
Communicated by: Dennis A. Hejhal
Copyright of article: Copyright 2000, American Mathematical Society

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