AI 中文总结
该研究针对标准格$\boldsymbol{\text{Z}}^d$上的本征函数，得到了三维下支撑计数的最优下界，改进了高维支撑维数界，还构造了稀疏调和函数，所有证明借助OpenAI Codex等工具完成并核验。
AI 中文摘要
我们研究标准格$\boldsymbol{\text{Z}}^d$上本征函数的稀疏支撑。对每个$d\boldsymbol{\text{≥}}3$，任何满足$u(0)\boldsymbol{\text{≠}}0$的实调和函数都满足$|\text{supp}(u)\boldsymbol{\text{∩}}Q_n^{(d)}|\boldsymbol{\text{≥}}\frac{10^{-10}}{d}\boldsymbol{\text{·}}n^2\boldsymbol{\text{（}}n\boldsymbol{\text{≥}}1\boldsymbol{\text{）}}$，该$n^2$阶在三维中是最优的。在零势情形下，这消除了Li和Zhang[Duke Math. J. 171 (2022), 327--415]支撑计数估计中的对数损失；在高维中，我们仅基于支撑的估计改进了Krymskii的支撑维数界[arXiv:2401.02800]，对$d\boldsymbol{\text{≥}}17$得到了超过2的指数，且趋近于$\text{log}_2d\boldsymbol{\text{-}}4$。我们还构造了稀疏调和函数，并确定了格本征函数全支撑的Zariski维数的最优下界。所有证明均通过OpenAI Codex、Ultra模式下的GPT-5.6 Sol完成，并由作者核验。
英文摘要
We study how sparsely a nonzero discrete harmonic function on the standard lattice $\mathbb{Z}^d$ can be supported. Let $Q_n^{(d)}=\{-n,\cdots,n\}^d$, and let $m_d(n)$ denote the least possible value of $|\mathrm{supp}(u)\cap Q_n^{(d)}|$ among discrete harmonic functions $u:\mathbb{Z}^d\to\mathbb{C}$ with $u(0)\neq0$. For all $n\geq1$, we prove \begin{equation*} m_3(n)\asymp n^2, \quad c_dn^{d^2/(2d-1)} \leq m_d(n)\leq (2n+1)^{\lfloor d/2\rfloor+1} \quad d\ge 4. \end{equation*} These estimates extend the two-dimensional support estimate of Buhovsky, Logunov, Malinnikova, and Sodin [Duke Math. J. 171 (2022), 1349--1378] to higher dimensions and obtain sharpness in dimension three. For $d\geq4$, the lower exponent and the upper one differ by less than $3/4$ in even dimensions and $1/4$ in odd dimensions. The proof combines Hilbert functions of finite support sets with a position-translation uncertainty principle. The sharp three-dimensional bound additionally uses Cayley--Bacharach relations and rigidity of algebraic curves. Finally, for every nonzero lattice eigenfunction with eigenvalue $λ$, the Zariski closure of its full support has dimension at least $\lceil d/2\rceil$, and at least $\lfloor d/2\rfloor+1$ when $λ\neq0$. Both bounds are optimal. All proofs resulted from human-guided exploration by GPT-5.6 Sol in Ultra mode and checked by the author.
Comments33 pages. The previous multiscale and endpoint collision arguments in the first version have been replaced by a nearly optimal Hilbert function approach. Material included only for proofreading has been removed
