Type: Article

Multiplicative congruences with variables from short intervals

Bourgain, J., Garaev, M. Z., Konyagin, S. V., Shparlinski, I. E.;

Abstract:

Recently, several bounds have been obtained on the number of solutions of congruences of the type where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval.