Glasnik matematički, Vol. 44 No. 1, 2009.
Original scientific paper
https://doi.org/10.3336/gm.44.1.02
On the distribution of solutions to linear equations
Igor Shparlinski
; Department of Computing, Macquarie University, Sydney, Australia
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.
Keywords
Linear equations; uniform distribution; exponential sums
Hrčak ID:
36943
URI
Publication date:
21.5.2009.
Visits: 1.077 *