Research proof index

If a finite set $s$ is additively dissociated, then its order-$n$ additive energy is at most $C^n n^n |s|^n$, where $C$ is the project's Chang constant. This is the quantitative dissociated-set estimate used in the proof of Chang's lemma.

An $L^\infty$ almost-periodicity theorem. Under the small-growth hypothesis $\sigma[A,S]\le K$, and for nonempty finite sets $B,C$, there is a set of translations $T$ with $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil/\varepsilon^2}|S|$. Every $t\in T$ changes the normalized convolution $\mu_A*1_B*\mu_C$ by at most $\varepsilon$ in $L^\infty$.

A boosted $L^\infty$ almost-periodicity estimate. Under $\sigma[A,S]\le K$, it finds a large set $T$, with the stated lower bound $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil k^2/\varepsilon^2}|S|$, such that averaging the target convolution against the $k$-fold convolution of $\mu_T$ changes it by at most $\varepsilon$ in $L^\infty$.

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 $\varepsilon$.

An approximate-homomorphism theorem for finite elementary abelian $2$-groups. Let $f:G\to G'$ and $K>0$. If at least a proportion $K^{-1}$ of pairs $(x,y)\in G^2$ satisfy $f(x+y)=f(x)+f(y)$, then there are an additive homomorphism $\varphi:G\to G'$ and a constant $c\in G'$ such that $f(x)=\varphi(x)+c$ for at least $|G|/(2^{144}K^{122})$ values of $x$.

If $A \subset {\bf F}_2^n$ is finite non-empty with $|A+A| \leq K|A|$, then there exists a subgroup $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$ such that $A$ can be covered by at most $2K^9$ translates of $H$.