The Maximum Number of Three Term Arithmetic Progressions, and Triangles in Cayley Graphs

Let $G$ be a finite Abelian group. For a subset $S\subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $=\frac{1}{\mathrm{\mid }S{\mathrm{\mid }}^2}{\sum }_{x,y\in S}{1}_S(x+y)$. For any $q\ge 1$ and $\alpha \in [0,1]$,andany$S \subseteq G$with$|S| = \frac{|G|}{q+\alpha}$,weshow$\frac{T_3(S)}{|S|^2}$andProb[$S$]areboundedaboveby$\max\left(\frac{q^2-\alpha q+\alpha^2}{q^2},\frac{q^2+2\alpha q+4\alpha^2-6\alpha+3}{(q+1)^2},\gamma_0\right)$,where$\gamma_0 < 1$ is an absolute constant. As a consequence, we verify a graph theoretic conjecture of Gan, Loh, and Sudakov for Cayley graphs.