Research proof index

A finite-field approximation lemma. If $A_1$ and $A_2$ each have density at least $\alpha$, then for any test set $S$ and $0<\varepsilon\le1$ there is a subspace $V$ of explicitly bounded codimension such that smoothing $\mu_{A_1}*\mu_{A_2}$ by the uniform measure on $V$ changes its total mass on $S$ by at most $\varepsilon$.

A finite-field density-increment lemma. If the normalized additive correlation of $A$ with a set $C$ of density at least $\gamma$ differs from its random value by at least $\varepsilon$, then there is a subspace $V$ of explicitly bounded codimension. Averaging $1_A$ over $V$ raises its $L^\infty$ density to at least $(1+\varepsilon/32)\alpha$, where $\alpha$ is the density of $A$.

Let $G$ be a finite vector space over $\mathbb F_q$, where $q\ge3$ is prime. If a nonempty set $A\subseteq G$ contains no nontrivial three-term arithmetic progression, then $\dim_{\mathbb F_q}G \le 2^{148}\,\mathcal L(\alpha)^9$, where $\alpha=|A|/|G|$ and $\mathcal L$ is the project's capped logarithm.