The distribution of rational points and polynomial maps on an affine variety over a finite field on average

Let $p$ be a prime, let $V/\mathbb{F}_p$ be an absolutely irreducible affine variety inside the affine $r$-space. In this paper, we consider the problem of how often a box $B$ will contain the expected number of points. In particular, we give a lower bound on the volume of $B$ that guarantees almost all translations of $B$ in the $r$-space contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense.