On small densities defined without pseudorandomness

We identify an assumption on linear forms ${\phi }_1,\dots ,{\phi }_k:{\mathbb{F}}_p^n\to {\mathbb{F}}_p$ that is much weaker than approximate joint equidistribution on the Boolean cube $\{0,1{\}}^n$ and is in a sense almost as weak as linear independence, but which guarantees that every subset of $\{0,1{\}}^n$ on which none of ${\phi }_1,\dots ,{\phi }_k$ has full image has a density which tends to 0 with $k$. This density is at most quasipolynomially small in $k$, a bound that is necessarily close to sharp.