Lattice balls with large additive energy in discrete cubes

Xinyu Long

For a finite set $A$ in an abelian group, let \[ E(A)=\#\{(a_1,a_2,a_3,a_4)\in A^4:a_1+a_2=a_3+a_4\}. \] We obtain an estimate uniform in $d$ that compares the normalized additive…