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Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity

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Pith's one-line read A nonzero real harmonic function on $\mathbb{Z}^d$ must have at least $c_d n^2$ supported points in every $n$-cube.

desk verdict A broad, ambitious paper with real new ideas and several likely sound results, but the proof of the main quadratic theorem has a concrete error in Lemma 2.6 and the headline result is not established. read the letter →

arxiv 2608.01673 v1 pith:OGHPHK56 submitted 2026-08-03 math.CA

Yunlei Wang This is my paper · ORCID
classification math.CA MSC 42B3739A1439A2214N05
keywords discreteharmonicfunctionslatticeeigenfunctionssupportgrowthquantitativeuniquecontinuationZariskidimensionCayley-BacharachpropertyHilbertfunctionmultiscalepacking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that real harmonic functions on the integer lattice $\mathbb{Z}^d$ cannot be sparse: for every $d\ge 3$, any harmonic $u$ with $u(0)\ne 0$ has at least $(10^{-10}/d)n^2$ supported points in the cube $\{-n,\ldots,n\}^d$ for every $n\ge 1$. In dimension three this quadratic growth is sharp, settling the finite-scale support-growth problem there and removing a logarithmic loss from the previous best three-dimensional bound. In high dimensions the paper's support-only arguments push the exponent above $2$ for $d\ge 17$ and within $4+o(1)$ of $\log_2 d$. The same framework yields sharp lower bounds on the Zariski dimension of the full support of a lattice eigenfunction. The governing idea is algebraic rigidity: a local support that is too thin must lie on a low-degree algebraic curve, and the directional structure of the lattice equation rules such curves out.

What carries the argument

The argument is carried by three interlocking mechanisms. First, the constant-coefficient operator $P=A_d-2d$ yields moment identities: any translate $v$ of $u$ satisfies $P^{m+1}(qv)=0$ when $\deg q\le m$, so on each local support $E(a,R)$ there is a linear dependence with nonzero coefficients, giving the Cayley–Bacharach property $CB(R-1)$ and, from independent translates, a Hilbert-rank bound $H_E(D-1)\le |E|-|F|$, where $H_E(s)$ is the rank of polynomial evaluations of degree at most $s$ on $E$. Second, a curve-carrier lemma converts these two facts into geometry: a finite set satisfying $CB(t)$ with Hilbert rank at most $20Dm$ lies on a reduced algebraic curve of degree at most $160m$, via the extremal Hilbert-function plateau theorem and elimination of isolated points. Third, a directional port rule — evaluating the equation at a zero neighbor forces a supported point among the displacements $e_i,2e_i,e_i\pm e_j$ — contradicts the existence of such a curve through Bézout intersection bounds and a component-cycle dimension argument. A separate support-only strand uses multiscale packing recurrences and endpoint-collision counts for long port words to deliver the superquadratic exponents in high dimensions.

What would settle it

Exhibit, for some $d\ge 3$, a harmonic function $u$ with $u(0)\ne 0$ and a sequence $n_k\to\infty$ such that $|\mathrm{supp}(u)\cap Q_{n_k}^{(d)}|\le C n_k^{2-\varepsilon}$; the theorem asserts no such function exists. A more targeted check is to construct a finite $Z\subset\mathbb{C}^d$ satisfying $CB(t)$ and the Hilbert-rank bound but lying on no curve of degree at most $160m$, which would falsify Lemma 2.6 directly.

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Extended reading notes

Core claim

The central claim is a quantitative rigidity theorem for lattice eigenfunctions. For every $d\ge 3$, the counting function $m_d(n)$ — the minimum number of support points a nonzero real harmonic function with $u(0)\ne 0$ must have in the cube $Q_n^{(d)}$ — satisfies $m_d(n)\ge 10^{-10}n^2/d$ for all $n\ge 1$, and in dimension three this order is optimal: the explicit product of an alternating-line function with an exponential factor is harmonic and has exactly $(2n+1)^2$ support points in $Q_n^{(3)}$. The same theorem gives $m_d(n)\gtrsim n^{\Theta_d}$ with $\Theta_d>2$ for $d\ge 17$ and $\Theta_d\ge \log_2 d-4-O(\log d/d)$ as $d\to\infty$, while a sparse product construction caps every $m_d(n)$ by $(2n+1)^{\lfloor d/2\rfloor+1}$. Separately, the Zariski closure of the support of any nonzero eigenfunction has dimension at least $\lceil d/2\rceil$, and at least $\lfloor d/2\rfloor+1$ when the eigenvalue is nonzero; both bounds are attained by the constructed products. The paper also proves support-only growth estimates that hold for every supportive set, at every center and every radius.

Load-bearing premise

The load-bearing premise is the curve-carrier lemma: a finite set of lattice points that satisfies the polynomial-interpolation condition inherited from the harmonic equation and has small Hilbert rank must lie on a reduced algebraic curve of degree at most $160m$; if that geometric reduction fails, the quadratic lower bound does not follow.

Editorial extensions

If this is right

  • In dimension three the support-growth problem is closed: $m_3(n)\asymp n^2$, with the logarithmic loss of the previous estimate removed and the quadratic order shown sharp.
  • Every dimension $d\ge 3$ gets a uniform quadratic floor for exact support in finite cubes; for $d\ge 17$ the bound is superquadratic, with exponent $\Theta_d\ge \log_2 d-4-o(1)$.
  • The support-only estimates apply to every supportive set, hence to supports of discrete Schrödinger solutions over any field, giving $|X\cap Q_N(x)|\ge c_{d,k}N^{\alpha_{d,k}}$ at every center and every radius.
  • The Zariski dimension of the support closure of a nonzero eigenfunction is at least $\lceil d/2\rceil$ (and $\lfloor d/2\rfloor+1$ for nonzero eigenvalue), and both bounds are attained by explicit lattice products.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The curve-carrier mechanism is not obviously confined to the zero-potential equation: a similar degree-lowering identity for bounded-potential operators would let the same curve contradiction remove the logarithmic loss for variable potentials, a case the paper explicitly does not treat.
  • The sharp Zariski bound plus the gap between $\Theta_d$ and $\lfloor d/2\rfloor+1$ suggests that closing Conjecture 1.4 needs a quantitative version of the component-cycle argument, since the paper itself notes Zariski density carries no uniform finite-scale information.
  • The multiscale packing and endpoint-collision exponents are natural targets for improvement; the upper constructions point toward $\lfloor d/2\rfloor+1$ as the true finite-scale exponent in every dimension, as the paper conjectures.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies the minimal local support size of nonzero real harmonic functions on Z^d, and more generally of eigenfunctions of the discrete adjacency operator. Its main theorem (Theorem 1.1) asserts that for every d ≥ 3 every real harmonic function with u(0) ≠ 0 satisfies |supp(u) ∩ Q_n^{(d)}| ≥ (10^{-10}/d) n^2, that the order n^2 is sharp in dimension three, and that for d ≥ 17 a superquadratic exponent can be obtained from support-only assumptions. The proof combines moment identities and Hilbert-function bounds to force sparse supports onto low-degree algebraic curves, a thin-shell selection, and two combinatorial estimates for supportive sets. The paper also proves sharp Zariski-dimension bounds for full supports in Theorem 1.3 and gives explicit sparse product constructions.

Significance. If the proof is correct, the paper settles the three-dimensional support-growth problem with the sharp exponent n^2, removes the logarithmic loss in the exact-support estimate of Li and Zhang, and improves Krymskii's support-dimension exponents at all finite scales. The explicit constructions, the detailed auxiliary lemmas, and the clean Zariski-dimension theorem are genuine strengths. However, the central quadratic bound rests on Lemma 2.6, whose proof in Appendix A.1 contains an invalid Macaulay-successor identity; without a correct proof of that lemma the main theorem and Corollary 1.2 are not established.

major comments (1)
  1. [Appendix A.1, Eq. (A.3)] The proof of Lemma 2.6 asserts that for an integer a with 0 ≤ a ≤ j, its j-th Macaulay successor satisfies a^{<j>} = a, and uses this to pass from Macaulay's inequality h_{j+1} ≤ h_j^{<j>} to the monotonicity h_{j+1} ≤ h_j. This identity is false: for j = 5 and a = 1, the unique Macaulay expansion is 1 = binom(5,5), so 1^{<5>} = binom(6,5) = 6, not 1. Consequently the monotonicity (A.3) is unsupported, and the subsequent pigeonhole argument producing s with h_s = h_{s+1} = e is invalid; a positive integer sequence bounded by 160m over more than 160m steps can alternate 1,2,1,2,... and have no consecutive equal terms. The invocation of the Bigatti–Geramita–Migliore plateau theorem (Theorem A.1) therefore does not follow, and Lemma 2.6 is not proved. Since Lemma 2.6 is the load-bearing curve-carrier step used in §3.2 to reach Proposition 3.5 and the quadratic lower bound, Theorem 1.1 and Corollary 1.2 are not established by the present proof. The authors must either correct this argument or supply a genuinely different proof of the plateau and of Lemma 2.6.
minor comments (4)
  1. [Title page and headers] The running title contains typographical artifacts: 'SP ARSE SUPPOR TS' should read 'SPARSE SUPPORTS', and 'LA TTICE' should read 'LATTICE'.
  2. [Figure 4 caption] The caption contains the typo 'fromed' instead of 'formed'.
  3. [Section 2.1, text after Proposition 2.1] The sentence 'the construction in Proposition 2.1 have the same order' should read 'the construction in Proposition 2.1 has the same order'.
  4. [Section 4.1, after Eq. (4.3)] The phrase 'its root against the value d is larger than two exactly when d > 16' is correct but would be clearer as 'exactly when d ≥ 17'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main lower-bound proofs are self-contained reductions using external algebraic-geometry theorems, not fits or self-citations.

full rationale

The paper's central derivation is not circular. The exponents in (1.2) and (1.3) are defined by explicit equations before any growth estimate is proved, and Lemma 4.4 then derives the growth bound from the packing recurrence by induction; the exponent is the solution of a^rho = s^rho + b, not a fitted quantity. The quadratic lower bound is obtained by contradiction: assuming a small support count yields a shell bound, the Hilbert-rank estimate (2.6), the curve-carrier Lemma 2.6 (whose proof invokes the external Bigatti-Geramita-Migliore theorem), and the curve-contradiction Proposition 3.5 lead to delta < r/(8d) and a shell-size threshold that contradict the hypotheses. Every inequality is displayed, and no 'predicted' quantity is a renamed input. The upper constructions in Proposition 2.1 are explicit products with a direct verification of harmonicity and support size. Comparisons with Li-Zhang and Krymskii are benchmarks, not inputs to the proofs. There are no self-citations and no uniqueness theorem imported from the author's prior work. The generative-AI disclosure concerns provenance, not circularity. The appendix claim that a^{<j>}=a for a <= j would, if incorrect, be a mathematical error in the proof of Lemma 2.6, not circular reasoning: an erroneous step is not the same as reducing a conclusion to its own assumptions. Under the stated circularity criteria, the derivation chain is self-contained.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims are derived from the lattice equation using explicit inequalities and standard algebraic geometry; no free parameters are fitted to data and no new physical entities are introduced.

assumptions (5)
  • standard math Bigatti-Geramita-Migliore plateau theorem
    Used in Lemma 2.6 to conclude a plateau in the Hilbert function forces a one-dimensional reduced scheme containing the finite set; cited from [2,20].
  • standard math Cayley-Bacharach property and related interpolation facts
    Definition 2.3 and Lemmas 3.1 and 3.2 use CB-type statements; cited from [5,6].
  • standard math Generic projection theorem and Bezout bounds for plane curves
    Used in Lemma 3.1, Lemma 3.2, and Proposition 3.5; cited from Harris [8] and Hartshorne [9].
  • standard math Serre duality for Cohen-Macaulay curves
    Used in Appendix A.2 to prove the interpolation bound on an irreducible curve.
  • domain assumption Discrete harmonic equation A_d u = 2d u as the definition of harmonic; support of a harmonic function satisfies the cross condition
    This is the mathematical setting of the paper, not an ad hoc assumption; the cross condition is proved from the equation.

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Pith. "Pith review of Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity." pith.science (2026). https://pith.science/paper/OGHPHK56

@misc{pith2026260801673,
  author       = {Pith},
  title        = {Pith review of: Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGHPHK56}},
  note         = {Machine review of arXiv:2608.01673}
}
abstract

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.

Figures

Figures reproduced from arXiv: 2608.01673 by the authors.

Figure 1
Figure 1. Numerically computed contributions to Θd for 2 ≤ d ≤ 2 14; their upper envelope is Θd. In dimension three, the quadratic exponent in Theorem 1.1 is sharp since the construction in Proposition 2.1 have the same order. We state it as a separate corollary. Corollary 1.2. Let u : Z 3 → R be harmonic and suppose that u(0) 6= 0. Then, for every integer n ≥ 1, |supp(u) ∩ Q (3) n | ≥ 10−10 3 n 2 . Proof. Apply the explicit … view at source ↗
Figure 2
Figure 2. The inner ball Br−D, the shell Br \ Br−D, and a translated ball a + BD ⊂ Br. The next lemma converts a Hilbert-rank bound into containment in an algebraic curve of controlled degree. Lemma 2.6. Let m ≥ 1. Let D be even and satisfy 104m ≤ D < 104m + 2. Suppose that a finite set Z ⊂ C d satisfies CB(t) for some t ≥ D and HZ(D − 1) ≤ 20Dm. Then Z is contained in a reduced algebraic curve of degree at most 160m. Detaile… view at source ↗
Figure 3
Figure 3. The local port rule. If p + ei is zero, the nonzero value arriving from p must be cancelled by at least one of the three marked candidate neigh￾bors. The following proposition turns forward translations among a finite family of irreducible varieties into translation symmetries and a dimension lower bound. Proposition 3.4. Let d ≥ 1, and let C be a finite nonempty family of irreducible affine subvarieties of A d C , … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: An example of a cycle fromed by V2, V3 and V4. We next pass from the discrete stabilizers to their complex span. Let F be a polynomial vanishing on V , and fix x ∈ V . By (3.10), the points x + nvi belong to V for every n ∈ Z. Therefore the one-variable polynomial z 7−…
Figure 5
Figure 5. Figure 5: A two-dimensional schematic of the packing behind (4.1). The endpoint cubes are mutually disjoint and also disjoint from a larger central cube; the outer square represents Qr+R+1(x). Lemma 4.4. Let a > s > 1, b > 0, and let F : Z≥0 → [1, ∞) be nondecreasing. Assume F(b…

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