SYSTEMS OF CONGRUENCES WITH PRODUCTS OF VARIABLES FROM SHORT INTERVALS

Abstract

We obtain an upper bound for the number of solutions to the system of $m$ congruences of the type $$\begin{eqnarray}\displaystyle \mathop{\prod }_{i=1}^{{\it\nu}}(x_{i}+s_{i})\equiv {\it\lambda}_{j}~(\text{mod }p)\quad j=1,\ldots ,m, & & \displaystyle \nonumber\end{eqnarray}$$ modulo a prime $p$, with variables $1\leq x_{i}\leq h$, $i=1,\ldots ,{\it\nu}$ and arbitrary integers $s_{j},{\it\lambda}_{j}$, $j=1,\ldots ,m$, for a parameter $h$ significantly smaller than $p$. We also mention some applications of this bound.