Common and Sidorenko equations in Abelian groups

A linear configuration is said to be common in a finite Abelian group G if for every 2-coloring of G the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation in an even number of variables over G , then it is common in F n p if and only if the equation’s coefficients can be partitioned into pairs that sum to zero mod p . This was proven by Fox, Pham and Zhao for sufficiently large n . We generalize their result to all sufficiently large Abelian groups G for which the equation’s coefficients are coprime to | G | .