On small densities defined without pseudorandomness

We identify a new sufficient condition on linear forms ${\varphi }_1,\dots ,{\varphi }_k:F_p^n\to F_p$ which guarantees that every subset of ${\{0,1\}}^n$ on which none of ${\varphi }_1,\dots ,{\varphi }_k$ has full image has a density which tends to 0 with k. The condition is much weaker than the condition usually used to guarantee that $\left({\varphi }_1\right(x),\dots ,{\varphi }_k(x\left)\right)$ takes each value of $F_p^k$ with probability close to $p^{-k}$ when x is chosen uniformly at random in the Boolean cube ${\{0,1\}}^n$. The density is at most quasipolynomially small in k, a bound that is necessarily close to sharp.