Common and Sidorenko Linear Equations

Abstract A linear equation with coefficients in $\mathbb{F}_q$ is common if the number of monochromatic solutions any two-coloring $\mathbb{F}_q^{\,n}$ asymptotically (as $n \to \infty$) at least expected a random two-coloring. The Sidorenko dense subset set same density. In this paper, we characterize those equations which are common, and Sidorenko. main novelty construction based on choosing Fourier that shows certain do not have these properties. This solves problems posed paper Saad Wolf.