Multiplicative Congruences with Variables from Short Intervals

Recently, several bounds have been obtained 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. 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 primes, one can always find an element of a large order in any rather short interval.