The distribution of solutions of the congruence $x_1x_2x_3\dots x_n\equiv c (\mod p)$

Abstract:

For a cube $\mathcal {B}$ of size $B$, we obtain a lower bound on $B$ so that $\mathcal {B}\cap V$ is nonempty, where $V$ is the algebraic subset of $\mathbb {F}_{p}^{n}$ defined by \begin{equation*}x_{1}x_{2}x_{3}\dots x_{n}\equiv c\pmod p ,\end{equation*} $n$ a positive integer and $c$ an integer not divisible by $p$. For $n=3$ we obtain that $\mathcal {B}\cap V$ is nonempty if $B\gg p^{\frac {2}{3}}(\log p)^{\frac {2}{3}}$, for $n=4$ we obtain that $\mathcal {B}\cap V$ is nonempty if $B\gg \sqrt {p}\log p$, and for $n\ge 5$ we obtain that $\mathcal {B}\cap V$ is nonempty if $B\gg p^{\frac {1}{4}+\frac {1}{\sqrt {2(n+4)}}}(\log p)^{\frac {3}{2}}$. Using the assumption of the Grand Riemann Hypothesis we obtain $\mathcal {B}\cap V$ is nonempty if $B\gg _{\epsilon }p^{\frac {2}{n}+\epsilon }$.