On a Ramsey--Turán variant of Roth's theorem

A classical theorem of Roth states that the maximum size of a solution-free set of a homogeneous linear equation $\mathcal{L}$ in $\mathbb{F}_p$ is $o(p)$ if and only if the sum of the coefficients of $\mathcal{L}$ is $0$. In this paper, we prove a Ramsey--Turán variant of Roth's theorem, with respect to a natural notion of ``structured'' sets introduced by Erdős and Sárközy in the 1970's. Namely, we show that the following statements are equivalent: $(a)$ Every solution-free set $A$ of $\mathcal{L}$ in $\mathbb{F}_p$ with $α(\mathrm{Cay}_{\mathbb{F}_p}(A)) = o(p)$ has size $o(p)$. $(b)$ There exists a non-empty \emph{subset} of coefficients of $\mathcal{L}$ with zero sum.