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An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

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Pith's one-line read Unusually high point probabilities for quadratic polynomials force near-low-rank algebraic structure, and this yields an optimal anti-concentration bound for edge counts in Ramsey graphs.

desk verdict A substantial paper: new algebraic inverse theorems for the quadratic Littlewood–Offord problem at near-1/n thresholds, backed by a genuinely new Fourier-decoupling technique and a clean Ramsey application; the proofs hold up. read the letter →

arxiv 1909.02089 v2 pith:6XNO65EH submitted 2019-09-04 math.CO math.PR

Matthew Kwan , Lisa Sauermann This is my paper · ORCID
classification math.COmath.PR MSC 05D4005C5560C05
keywords Littlewood-Offordproblemquadraticpolynomialsanti-concentrationinversetheoremslow-rankformsRamseygraphsedge-statisticsRademachervariables
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

This paper proves an inverse theorem for the quadratic Littlewood–Offord problem: if a quadratic polynomial in independent fair coin flips has an unusually high chance of taking one particular value, then the polynomial must be algebraically close to a low-rank quadratic form. The threshold is roughly $n^{-1}$, the natural scale for a quadratic with bounded coefficients, and the paper proves both an $\ell^1$-coefficient version and a Hamming-distance version when the quadratic coefficients lie in a finite set. If the theorem is right, unusually strong concentration cannot occur without an algebraic reason: the polynomial is forced to sit near a form whose symmetric coefficient matrix has rank less than $r$. The paper also uses the finite-coefficient version to show that in every $C$-Ramsey graph, the number of edges in a random $k$-vertex subset has point probabilities at most $n^{-1+o(1)}$, asymptotically answering a question from earlier work on edge-statistics.

What carries the argument

The engine is a decoupling trick applied to the characteristic function of $f(\xi)$: splitting the variables into two groups and subtracting an independent copy turns the quadratic polynomial into a linear one, with the usual square-root loss moved inside an integral where a sharp threshold in $|t|$ makes it negligible. The linear analysis then uses a multidimensional Littlewood–Offord theorem for robustly non-degenerate matrices, and Esséen's inequality converts the resulting Fourier decay into point-probability bounds. A separate layer of robust linear independence ($\varepsilon$-independence) and a symmetric low-rank approximation lemma turn the condition 'far from a low-rank quadratic form' into the non-degeneracy that the Fourier argument requires. Here the rank of a quadratic form is the rank of its symmetric coefficient matrix, equivalently the minimum number of squares of linear forms needed to express it.

What would settle it

A sequence of quadratic polynomials with all coefficients in $\{-1,0,1\}$ whose exact point probabilities exceed $C(\log n)^{r/2}n^{-1+2/(r+2)}$ for some fixed $r\ge3$, while their coefficient matrices stay at $\ell^1$-distance at least $\varepsilon n^2$ from every symmetric matrix of rank less than $r$, would refute Theorems 1.1 and 1.2. A concrete starting point is exhaustive search over small $n$ with coefficients in $\{0,1\}$, computing both the maximum point probability and the minimal $\ell^1$ distance to low-rank symmetric matrices.

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Extended reading notes

Core claim

The central claim is a rigidity principle: for a quadratic polynomial $f\in F[x_1,\ldots,x_n]$ with all coefficients of absolute value at most $1$, if $\sup_x\Pr(f(\xi)=x)\ge C(\log n)^{r/2}n^{-1+2/(r+2)}$, then there is a quadratic form $h$ of rank strictly less than $r$ whose coefficients differ from those of $f$ by total absolute value at most $\varepsilon n^2$. When the degree-$2$ coefficients of $f$ are drawn from a finite set, the same concentration forces $f$ and $h$ to differ in at most $\varepsilon n^2$ coefficients. The statements hold over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{Q}$, and in the finite-coefficient case $h$ can be chosen with coefficients in a finite set depending only on $r$ and the allowed coefficients. For the Ramsey application, the random edge count is represented as a quadratic polynomial whose coefficient matrix contains many full-rank $r\times r$ submatrices, so it is far from every low-rank quadratic form, and Theorem 1.2 delivers the bound.

Load-bearing premise

The application to Ramsey graphs relies on the imported graph counting estimate that every $C$-Ramsey graph contains induced copies of every small fixed graph in numbers proportional to $n^h$; if that estimate fails with a full polynomial factor, the deduction of Theorem 1.3 breaks down, although the inverse theorems themselves do not depend on it.

Editorial extensions

If this is right

  • A quadratic polynomial with bounded coefficients whose point probabilities exceed $n^{-1+o(1)}$ must be within $\varepsilon n^2$ coefficient-wise of a low-rank quadratic form, so strong concentration is a certificate of algebraic degeneracy.
  • With $r=3$, point probabilities much larger than $n^{-3/5}$ force the polynomial to be close to one that splits into linear factors over the complex numbers, the qualitative picture predicted by the earlier conjecture.
  • Every $C$-Ramsey graph has the property that the edge count of a uniformly random subset of size between $cn$ and $(1-c)n$ has point probabilities at most $n^{-1+o(1)}$, matching the random-graph benchmark up to the $o(1)$ term.
  • The finite-coefficient Hamming version gives a template for proving anti-concentration of graph statistics: exhibit many disjoint full-rank submatrices in the associated coefficient matrix, then invoke the inverse theorem.

Reading between the lines

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

  • The two distance notions invite a computational test: for a candidate statistic, one can compute the robust rank of its coefficient matrix and compare the implied concentration bound with the true point probabilities, which could expose where the $\ell^1$ approximation is too crude.
  • The proof does not obviously extend to degree $3$, since iterating the decoupling gives weaker control; testing the method on cubic forms with tensor-rank-type structure would show whether the low-rank conclusion is special to quadratics.
  • Removing the $o(1)$ in the Ramsey-graph bound would make the edge-count distribution exactly match the random-graph benchmark; a plausible route is to sharpen the symmetric low-rank approximation lemma from aggregate $\ell^1$ control to control on almost every entry.
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Desk editor's note, referee report, and a circularity audit.

Referee Report

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Summary. The paper proves algebraic inverse theorems for the quadratic Littlewood–Offord problem. Theorem 1.1 states that if a quadratic polynomial f over F∈{C,R,Q} with coefficients of modulus at most 1 has point probability at least C(log n)^{r/2}/n^{1-2/(r+2)} for some r≥3, then f is within coefficient L1 distance εn^2 of a quadratic form of rank strictly less than r. Theorem 1.2 gives the analogous Hamming-distance conclusion when the degree-2 coefficients lie in a finite set S. Theorem 1.3 applies Theorem 1.2 to show that in any C-Ramsey graph, the number X of edges induced by a uniformly random k-vertex subset, with k=Θ(n), satisfies Pr(X=x)≤n^{-1+o(1)}. The proof strategy is a Fourier-decoupling anti-concentration lemma (Lemma 3.2), a complex-to-real reduction via random phases (Lemma 4.1), and a substantial development of robust linear-independence tools (Sections 6–7) culminating in symmetric low-rank approximation statements (Lemmas 5.5 and 5.7).

Significance. If correct, these are the first inverse theorems of this strength for quadratic Littlewood–Offord problems, giving structural conclusions at concentration levels well above the 1/√n barrier and making significant progress toward Costello's conjecture in the algebraic direction. The technical apparatus—decoupling inside the characteristic function, robust linear independence, and simultaneous symmetric low-rank approximation—is likely to be independently useful. The Ramsey application is a clean and nontrivial consequence. I explicitly note that the central inverse theorems (Theorems 1.1 and 1.2) do not rely on Lemma 2.2; that lemma is used only for the Ramsey application and is a standard regularity-method counting statement. The proofs of the key technical lemmas are complete, and I found no load-bearing error in the central derivation.

minor comments (4)
  1. [§7.1, Claim 7.5] In the proof of Claim 7.5, the same expression ‖(A′−B′)I‖1 is written twice in consecutive displayed equations, and the identity ‖(H−A′)I‖1=‖(H−A′)I‖1 also appears twice; one member of each pair should refer to the column-indexed submatrix rather than the row-indexed submatrix. Please clarify the notation distinguishing row and column submatrices in this claim.
  2. [§4, Lemma 4.1] In the proof of Lemma 4.1, the definition of p(z) and the displayed formula for det Re(e^{iθ}A) should involve the conjugate matrix \overline{A} rather than A, since Re(e^{iθ}A)=(e^{iθ}A+e^{-iθ}\overline{A})/2. As written, both displays are missing the conjugation.
  3. [§3, proof of Lemma 3.2] In the final Esséen integration, the limits of the first integral are written from 1/√n to 1/n but should be from 1/n to 1/√n, and the outer integral should read from −1/s to 1/s rather than from 1/s to −1/s. The numerical evaluation is correct under the intended limits.
  4. [Throughout] Several exponents such as α(6r)^{q−r} and α(6r)^{−r} are ambiguous in the typeset text and can be misread as products rather than nested powers; using explicit braces, e.g., α^{(6r)^{q−r}}, would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inverse theorems are derived from external anti-concentration inputs and self-contained algebraic lemmas, and the Ramsey application rests on a standard regularity counting lemma.

full rationale

Theorems 1.1 and 1.2 are proved by contrapositive: if the coefficient matrix has many robustly independent rows, Lemma 3.2 gives the upper bound O((log n)^{r/2}/n^{1-2/(r+2)}) for every point probability; otherwise Lemmas 5.5 and 5.7 produce a low-rank quadratic form close to f. Lemma 3.2 is proved from Halasz's multidimensional Littlewood-Offord theorem (Theorem 3.5) and Esseen's concentration inequality (Lemma 3.4), both external results, together with a decoupling argument (Lemma 3.3) that is fully proved in the text. The robust linear independence lemmas (Sections 6-7) are proved from elementary linear algebra and determinant estimates, not from the statements of Theorems 1.1 or 1.2. No parameter is fitted to the target conclusion, and no point-probability bound is assumed in deriving the structural conclusion. The sole external graph-theoretic input, Lemma 2.2 (many induced copies of every small graph in Ramsey graphs), is cited to [10] and [19] and is a standard consequence of Szemeredi regularity; it is used only in the application Theorem 1.3 via Claim 2.4, not in Theorems 1.1 or 1.2. The coupling representation (Lemma 2.3) is described as a variant of a lemma in [27], but it is stated and proved in full in the paper, so the self-citation is not load-bearing. There is no renamed known result, no uniqueness imported from the authors' prior work, and no ansatz smuggled in by self-citation. Accordingly, the derivation chain is self-contained and no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard theorems from the literature, none authored by the present authors. No free parameters fitted to data and no new postulated entities appear in the paper.

assumptions (4)
  • standard math Halasz's multi-dimensional Littlewood-Offord inequality (Theorem 3.5)
    Invoked in Claim 3.7 to bound the probability that several linear coefficient combinations fall near lattice points; cited to Halasz [24]. Essential for the core anti-concentration Lemma 3.2.
  • standard math Esseen's concentration inequality (Lemma 3.4)
    Used to convert bounds on the characteristic function into an anti-concentration bound on the random variable; cited to Tao and Vu [46, Lemma 7.17].
  • domain assumption Szemeredi regularity lemma and counting lemmas (Lemma 2.2, cited to [10] and [19])
    Used to prove that every C-Ramsey graph contains Ω(n^h) induced copies of every h-vertex graph. This graph-theoretic counting result is load-bearing for the Ramsey application (Theorem 1.3).
  • standard math Standard concentration inequalities (Chernoff, Azuma-Hoeffding, McDiarmid)
    Used throughout for concentration of strong tuples (Claim 2.4) and for the random partition in Lemma 3.2.

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Pith. "Pith review of An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs." pith.science (2026). https://pith.science/paper/6XNO65EH

@misc{pith2026190902089,
  author       = {Pith},
  title        = {Pith review of: An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6XNO65EH}},
  note         = {Machine review of arXiv:1909.02089}
}
abstract

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger than $1/n$ then it must be close to a quadratic form with low rank. We also give an application to Ramsey graphs, asymptotically answering a question of Kwan, Sudakov and Tran.

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