Type: Article

Double exponential sums and congruences with intervals and exponential functions modulo a prime

Garaev, M. Z.;

Abstract:

Let p be a large prime number and g be any integer of multiplicative order T modulo p. We obtain a new estimate of the double exponential sum S=?n?N|?m?Me p (ang m )|,gcd?(a,p)=1, where N and M are intervals of consecutive integers with |N|=N and |M|=M?T elements. One representative example is the following consequence of the main result: if N=M?p 1/3 , then |S|<N 2?1/8+o(1) . We then apply our estimate to obtain new results on additive congruences involving intervals and exponential functions