Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity

2026/08/03 by Yunlei Wang
Mathematics · #math.CA #msc:42B37 #msc:39A14 #msc:39A22 #msc:14N05

paper · pdf

38 pages, 5 figures

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

We study sparse supports of eigenfunctions on the standard lattice ℤ^d. For every d≥3, any real harmonic function with u(0)≠0 satisfies |supp(u)∩ Q_n^(d)|≥ \frac10^-10d n² (n≥1). The order n² 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≥17 and approach log₂d-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.

Citations

• On the lowest possible dimension of supports of solutions to the discrete Schrodinger equation
• Anderson–Bernoulli localization on the three-dimensional lattice and discrete unique continuation principle
• A Cayley-Bacharach theorem and plane configurations
• Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice
• Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice
• A discrete harmonic function bounded on a large portion of ℤ² is constant
• A discrete harmonic function bounded on a large portion of Z2 is constant
• A Rellich Type Theorem for Discrete Schrödinger Operators
• The geometry of Hilbert functions
• Remarks on Divisors of Zero