On congruences involving product of variables from short intervals

Abstract: We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv 0\pmod p$, then the congruence $$ ax_1\cdots x_6+bx_7\cdots x_{13}\equiv c\pmod p $$ has a solution with $x_j\in\cI_j$. There exists an absolute constant $n_0\in\N$ such that for any $0<\varepsilon<1$ and any sufficiently large prime $p$, any quadratic residue $\lambda$ modulo $p$ can be represented in the form $$ x_1\cdots x_{n_0}\equiv \lambda\pmod p,\quad x_i\in\N,\quad x_i\le p^{1/(4e^{2/3})+\varepsilon}. $$ For any $\varepsilon>0$ there exists $n=n(\varepsilon)\in \N$ such that for any sufficiently large $m\in\N$ the congruence $$ x_1\cdots x_{n}\equiv 1\pmod m,\quad x_i\in\N,\quad x_i\le m^{\varepsilon} $$ has a solution with $x_1\not=1$.