离散立方体中具有大加性能的格球
Lattice balls with large additive energy in discrete cubes
AI总结：
本文研究离散立方体中格球的加性能，得到其归一化加性能的渐近估计，构造了满足特定性能条件的集合，回答了Shao的问题，并确定了连续球性能指数速率的极值格拉姆矩阵。
AI中文摘要：
对于阿贝尔群中的有限集合$A$，定义加性能$E(A)=\text{#}\bigl\{(a_1,a_2,a_3,a_4)\text{∈}A^4:a_1+a_2=a_3+a_4\bigr\}$。本文得到了关于维度$d$一致的估计，比较$\boldsymbol{Z}^d\boldsymbol{\text{∩}}B_d(R)$的归一化加性能与$B_d(R)$的连续性能：若$R_d/\boldsymbol{\text{√}}d\boldsymbol{\text{→}}\boldsymbol{\text{∞}}$，则$\boldsymbol{\text{lim}}_{d\boldsymbol{\text{→}}\boldsymbol{\text{∞}}}\bigl(\frac{E(\boldsymbol{Z}^d\boldsymbol{\text{∩}}B_d(R_d))}{|\boldsymbol{Z}^d\boldsymbol{\text{∩}}B_d(R_d)|^3}\bigr)^{1/d}=\frac{4\boldsymbol{\text{√}}3}{9}$。作为应用，取$A_n=R_n\boldsymbol{1}_d+(\boldsymbol{Z}^d\boldsymbol{\text{∩}}B_d(R_n))$，其中维度$d=d(n)\boldsymbol{\text{→}}\boldsymbol{\text{∞}}$且满足$\boldsymbol{\text{log}}d=o(\boldsymbol{\text{log}}n)$，$R_n=\boldsymbol{\text{⌊}}(n-1)/2\boldsymbol{\text{⌋}}$，此时$A_n\boldsymbol{\text{⊂}}\boldsymbol{\text{\textbraceleft}}0,1,\boldsymbol{\text{…}},n-1\boldsymbol{\text{\textbraceright}}^d$，且$\boldsymbol{\text{log}}E(A_n)=3\boldsymbol{\text{log}}|A_n|-d\boldsymbol{\text{log}}\frac{3\boldsymbol{\text{√}}3}{4}+o(d)$。特别地，取$d=\boldsymbol{\text{⌊}}(\boldsymbol{\text{log}}n)^{1/2}\boldsymbol{\text{⌋}}$可得到显式构造，回答了Shao提出的问题。本文还证明，在格拉姆矩阵坐标下，连续球性能的指数速率由固定维度的行列式最大化决定，其极值为正四面体的格拉姆矩阵。
英文摘要：
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.
