On congruences involving product of variables from short intervals

Abstract

We prove several results which imply the following consequences. For any $\epsilon >0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv 0(\mathrm{mod}p)$, then the congruence $$a{x}_1\cdots x_6+b{x}_7\cdots x_{13}\equiv c(\mathrm{mod}p)$$ has a solution with $x_j\in\cI_j$. There exists an absolute constant $n_0\in \mathbb{N}$ such that for any $0<\epsilon <1$ and any sufficiently large prime $p$, any quadratic residue $\lambda$ modulo $p$ can be represented in the form $$x_1\cdots x_{n_0}\equiv \lambda (\mathrm{mod}p),x_i\in \mathbb{N},x_i\le p^{1/(4e^{2/3})+\epsilon }.$$ For any $\epsilon >0$ there exists $n=n(\epsilon )\in \mathbb{N}$ such that for any sufficiently large $m\in \mathbb{N}$ the congruence $$x_1\cdots x_n\equiv 1(\mathrm{mod}m),x_i\in \mathbb{N},x_i\le m^{\epsilon }$$ has a solution with $x_1\ne 1$.