Generic solutions to symmetric linear equations

Abstract: In 1993, Ruzsa showed that for every $k \geq 2$, there exists a constant $C$ such that every subset $A \subseteq [N]$ of size at least $C N<sup>{1/k}$ contains $2k$ distinct elements $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ such that $a_1 + \cdots + a_k = b_1 + \cdots + b_k$. We strengthen this result by proving that the elements $a_1, \ldots, a_k, b_1, \ldots, b_k$ can be chosen to have the additional property that ${a_1, \ldots, a_k, b_1, \ldots, b_k}$ has $2<sup>{2k}-1$ distinct subset sums, with the only coincidence being that ${a_1, \ldots, a_k}$ and ${b_1, \ldots, b_k}$ have the same sum. Our proof also applies to any finite Abelian group of odd order $N$, and it provides a corresponding supersaturation result: that whenever $|A| \geq CN<sup>{1/k}$, there are at least $Ω(|A|<sup>{2k}/N)$ choices for $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ satisfying these properties. We prove a slightly weaker statement for Abelian groups of even order. We also apply our methods to the vector space setting and prove the following $\mathbb F_q$-analogue of the Bondy-Simonovits Theorem on the extremal number of even cycles in graphs: Any rank-$n$, simple, $\mathbb F_q$-representable matroid with no circuit of size exactly $2k$ has size at most $C q<sup>{n/k}$ for some constant $C$ depending only on $q$ and $k$. When $q=2$, this is best possible up to the constant $C$ for all $k \geq 2$. Our methods also apply to the original graph setting and give a new proof of the Bondy-Simonovits Theorem and its supersaturation version.