  We obtain upper bounds on the number of solutions to congruences of the type
$$ (x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p $$
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. B. Friedlander and H. Iwaniec and some results of M.-C. Chang
and A. A. Karatsuba on character sums twisted with the divisor function.
