Glasnik Matematicki, Vol. 44, No.1 (2009), 7-10.

ON THE DISTRIBUTION OF SOLUTIONS TO LINEAR EQUATIONS

Igor E. Shparlinski

Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
e-mail: igor@ics.mq.edu.au

Abstract.   Given two relatively prime positive integers m < n we consider the smallest positive solution (x0, y0) to the equation mx - ny = 1. E. I. Dinaburg and Y. G. Sinai have used continued fractions to show that the ratios x0/n are uniformly distributed in [0,1], when n and m run through consequtive integers of intervals of comparable sizes. We use a bound of exponential sums due to W. Duke, J. B. Friedlander and H. Iwaniec to show a similar result when m and n run through arbitrary sets which are not too thin.

2000 Mathematics Subject Classification.   11D04, 11K38, 11L40.

Key words and phrases.   Linear equations, uniform distribution, exponential sums.

Full text (PDF) (free access)

DOI: 10.3336/gm.44.1.02

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