#### 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 (*x*_{0}, *y*_{0})
to the equation *mx* - *ny* = 1. E. I. Dinaburg and Y. G. Sinai
have used continued fractions
to show that the ratios *x*_{0}/*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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