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.