Lattice balls with large additive energy in discrete cubes

For a finite set $A$ in an abelian group, let $$E(A)=\mathrm{\#}\{(a_1,a_2,a_3,a_4)\in A^4:a_1+a_2=a_3+a_4\}\mathrm{.}$$ 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 $R_d\mathrm{/}\sqrt{d}\to \mathrm{\infty }$, then $$\underset{d\to \mathrm{\infty }}{\lim }{\left(\frac{E({\mathbb{Z}}^d\cap B_d(R_d))}{\mid {\mathbb{Z}}^d\cap B_d(R_d){\mid }^3}\right)}^{1\mathrm{/}d}=\frac{4\sqrt{3}}{9}\mathrm{.}$$ As an application, consider $$A_n=R_n{\mathbf{1}}_d+({\mathbb{Z}}^d\cap B_d(R_n)),$$ where $d=d(n)\to \mathrm{\infty }$ satisfy $\log d=o(\log n)$, and $R_n=\lfloor (n-1)\mathrm{/}2\rfloor$. Then $A_n\subset \{0,1,\dots ,n-1{\}}^d$ and $$\log E(A_n)=3\log \mathrm{\mid }{A}_n\mathrm{\mid }-d\log \frac{3\sqrt{3}}{4}+o(d)\mathrm{.}$$ In particular, taking $d=\lfloor (\log n{)}^{1\mathrm{/}2}\rfloor$ 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.