Common and Sidorenko Linear Equations

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