A strengthening of Chang's lemma

We prove a strengthening of Chang's lemma for subsets of $\mathbb F_p^n$. The classical conclusion that the large spectrum is contained in a subspace of dimension at most $2\varepsilon^{-2}\log(1/\alpha)$ is refined to show that every character outside this subspace has small correlation with the set not only globally, but also on average over the cosets of the orthogonal complement, in a natural cosetwise $\ell^1$ norm. As a consequence, we obtain a localized counting lemma. We also give an extension of the argument to arbitrary finite abelian groups.