Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity

We study sparse supports of eigenfunctions on the standard lattice $\mathbb{Z}^d$. For every $d\ge3$, any real harmonic function with $u(0)\ne0$ satisfies $$|\mathrm{supp}(u)\cap Q_n^{(d)}|\ge \frac{10^{-10}}{d}\,n^2 \qquad(n\ge1).$$ The order $n^2$ is sharp in dimension three. In the zero-potential case, this removes the logarithmic loss in the support-count estimate of Li and Zhang [Duke Math. J. 171 (2022), 327--415]. In high dimensions, our support-only estimates improve Krymskii's support-dimension bound [arXiv:2401.02800], yielding exponents that exceed two for $d\ge17$ and approach $\log_2d-4$. We also construct sparse harmonic functions and determine the sharp lower bounds for the Zariski dimension of the full support of a lattice eigenfunction. All proofs were obtained through OpenAI Codex, GPT-5.6 Sol in Ultra mode, and checked by the author.