 Finite-field character approximation

Sampling the normalized Fourier expansion of $f:{\mathbb{F}}_p^n\to \mathbb{C}$ gives an average of $O(q/{ϵ}^2)$ phase-adjusted characters whose $L^q$ error is at most $ϵ\Vert \hat{f}{\Vert }_1$. The common kernel of those characters is therefore a low-codimension subspace of almost periods of $f$.