3 problems

Let $A$ be a subset of an abelian group $G$, and suppose that there are at least $b4|A|^3$ additive quadruples in $A^4$ satisfying $a_1+a_2=a_3+a_4$. A Bourgain system…

Let $A$ be a finite set with additive energy satisfying … Suppose there are subsets $A',B'\subseteq A$ with … and … where $c_2,c_3,c_4>0$ are constants. Optimal exponent conjecture…

Let $G$ be a locally compact abelian group, let $A\subset G$ be a compact neighborhood, and let $\mu$ be Haar measure on $G$. For $n\geqslant 1$, write $nA$ for the $n$-fold sumset…