Distribution of solutions to systems of congruences in balls

Let $G_1,\dots ,G_n\in {\mathbb{F}}_p[X_1,\dots ,X_m]$ be $n$ polynomials in $m$ variables over the finite field ${\mathbb{F}}_p$ of $p$ elements. For any sufficiently large prime $p$ and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of $G=(G_1,\dots ,G_n)$, we study various properties regarding the distribution of the vectors by fractional parts \begin{equation*} \bigg(\bigg{ \frac{G_1($\mathbf{\text{x}}$)}{p}\bigg},\cdots,\bigg{ \frac{G_n($\mathbf{\text{x}}$)}{p}\bigg}\bigg)\in $\mathbb{T}$^n,\hspace{10pt} $\mathbf{\text{x}}$\in $\mathbb{F}$_p^m. \end{equation*} We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.