On a theorem of Shkredov

We show that if \( A \) is a finite subset of an abelian group with additive energy at least \( c|A|^3 \), then there is a set \( \mathcal{L} \subset A \) with \( |\mathcal{L}| = O(c^{-1} \log |A|) \) such that \( |A \cap \mathrm{Span}(\mathcal{L})| = \Omega(c^{1/3} |A|) \).