Equidistribution of high-rank polynomials with variables restricted to subsets of ${\mathbb{F}}_p$

Let $p$ be a prime and let $S$ be a non-empty subset of ${\mathbb{F}}_p$. Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P:{\mathbb{F}}_p^n\to {\mathbb{F}}_p$ is a polynomial and its restriction to $S^n$ does not take each value with approximately the same frequency, then there exists a polynomial $P_0:{\mathbb{F}}_p^n\to {\mathbb{F}}_p$ that vanishes on $S^n$, such that the polynomial $P-P_0$ has bounded rank. Our argument uses two black boxes: that a tensor with high partition rank has high analytic rank and that a tensor with high essential partition rank has high disjoint partition rank.