Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity

[2608.01673] Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Classical Analysis and ODEs arXiv:2608.01673 (math) [Submitted on 3 Aug 2026 ( v1 ), last revised 14 Aug 2026 (this version, v2)] Title: Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity Authors: Yunlei Wang View a PDF of the paper titled Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity, by Yunlei Wang View PDF HTML (experimental) Abstract: 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 $\lambda$, the Zariski closure of its full support has dimension at least $\lceil d/2\rceil$, and at least $\lfloor d/2\rfloor+1$ when $\lambda\neq0$. Both bounds are optimal. All proofs resulted from human-guided exploration by GPT-5.6 Sol in Ultra mode and checked by the author. Comments: 33 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 The previous multiscale and endpoint collision arguments in the first version have been replaced by a nearly optimal Hilbert function approach.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 Subjects: Classical Analysis and ODEs (math.CA) MSC classes: 42B37 (Primary) 39A14, 39A22, 14N05 (Secondary) Cite as: arXiv:2608.01673 [math.CA] (or arXiv:2608.01673v2 [math.CA] for this version) https://doi.org/10.48550/arXiv.2608.01673 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yunlei Wang [ view email ] [v1] Mon, 3 Aug 2026 04:07:59 UTC (45 KB) [v2] Fri, 14 Aug 2026 20:55:04 UTC (34 KB) Full-text links: Access Paper: View a PDF of the paper titled Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity, by Yunlei Wang View PDF HTML (experimental) TeX Source view license Current browse context: math.CA < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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