Low-complexity approximations for sets defined by generalizations of affine conditions

Abstract Let p p $p$ be a prime, let upper S S $S$ be a non-empty subset of double struck upper F Subscript p F p $\mathbb{F}_p$ and let 0 less than epsilon less than or equals 1 0 < ϵ ≤ 1 $0\lt \epsilon \leq 1$ . We show that there exists a constant upper C equals upper C left parenthesis p comma epsilon right parenthesis C = C ( p , ϵ ) $C=C(p, \epsilon )$ such that for every positive integer k k $k$ , whenever phi 1 comma ellipsis comma phi Subscript k Baseline colon double struck upper F Subscript p Superscript n Baseline right arrow double struck upper F Subscript p Baseline ϕ 1 , … , ϕ k : F p n → F p $\phi _1, \dots , \phi _k: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ are linear forms and upper E 1 comma ellipsis comma upper E Subscript k Baseline