Lattice balls with large additive energy in discrete cubes

Abstract

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 energy of $\mathbb{Z}^d \cap B_d(R)$ with the continuous energy of $B_d(R)$ . If $R_d/\sqrt d\to\infty$, then \[ \lim_{d\to\infty} \left( \frac{E\bigl(\mathbb{Z}^d\cap B_d(R_d)\bigr)} {\lvert \mathbb{Z}^d\cap B_d(R_d)\rvert^3} \right)^{1/d} =\frac{4\sqrt{3}}{9}. \] As an application, consider \[ A_n = R_n\mathbf{1}_d + \bigl(\mathbb{Z}^d\cap B_d(R_n)\bigr), \] where $d=d(n)\to\infty$ satisfy $\log d=o(\log n)$, and $R_n=\lfloor(n-1)/2\rfloor$. Then $A_n\subset\{0,1,\ldots,n-1\}^d$ and \[ \log E(A_n) =3\log|A_n|-d\log\frac{3\sqrt3}{4}+o(d). \] In particular, taking $d=\lfloor(\log n)^{1/2}\rfloor$ gives an explicit construction answering a question of Shao \cite{Shao2026}. We also prove that in Gram-matrix coordinates, the exponential rate of the continuous ball energy is determined by a fixed dimensional determinant maximization whose extremizer is the Gram matrix of a regular tetrahedron.