On congruences with products of variables from short intervals and applications

We obtain upper bounds on the number of solutions to congruences of the type $(x_1+s)\dots(x_\nu+s)\equiv(y_1+s)\dots(y_\nu +s)\not\equiv0\pmod p$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function.