Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Algebraic aspects of the polynomial Littlewood-Offord problem

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves optimal anticoncentration for multilinear polynomials (point probabilities $n^{-1+\varepsilon}$ unless nearly reducible), power savings for complex quadratics, and disproves the original multilinear conjecture in degree…

desk verdict Real new results and a useful counterexample, but the appendix proof of the geometric Littlewood-Offord theorem has a counting error that breaks the self-contained proof of the main multilinear theorem. read the letter →

arxiv 2505.23335 v1 pith:ZPVJOPEB submitted 2025-05-29 math.CO math.NTmath.PR

Zhihan Jin , Matthew Kwan , Lisa Sauermann , Yiting Wang This is my paper
classification math.COmath.NTmath.PR MSC 05D4060E1515A6911B30
keywords Littlewood–OffordproblemanticoncentrationRademacherrandomvariablesmultilinearformspartitionrankquadraticpolynomialsinversetensorpropertytesting
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 studies how much a degree-$d$ polynomial of $n$ independent Rademacher variables can concentrate on a single value. It argues that the general $n^{-1/2}$-type bound can be dramatically improved unless the polynomial is algebraically structured, namely close to a reducible polynomial or to a low-rank quadratic form. As its flagship result, it shows that every $d$-multilinear form either is close to a product of lower-degree multilinear factors or has point probability at most $n^{-1+\varepsilon}$, resolving a repaired form of the multilinear conjecture; since examples from the multiplication table problem show no bound below $(\log n)^{\alpha}/n$ is possible, the $n^{-1}$ exponent is optimal up to $n^{\varepsilon}$. For complex quadratic polynomials that are robustly irreducible it proves a power saving with exponent $13/24$, and for general quadratics high rank of the quadratic part forces point probabilities near $n^{-1}$. An appendix disproves the original multilinear conjecture by exhibiting robustly irreducible degree-$d$ polynomials with point probability at least $\varepsilon/n$.

What carries the argument

The machinery has five linked components. First, a local-to-global tensor lemma: if all but a $\delta$-fraction of small $2^{d-1}\times\cdots\times 2^{d-1}$ subtensors of a $d$-dimensional tensor are reducible, then changing an $\varepsilon$-fraction of entries makes the whole tensor reducible, with $\delta=(\varepsilon/2)^{2^{d-1}}$; the symmetric-matrix analogue says many singular $r\times r$ submatrices force near-low-rank while preserving symmetry. Second, a classification of the maximal linear subspaces of the variety of reducible tensors: each is a fixed tensor $T^\star$ on one subset of the $d$ types, tensored with arbitrary tensors on the complementary types. Third, a geometric Littlewood–Offord theorem: a Rademacher sum of vectors lands in a proper variety with probability $\ge n^{-1/2+\varepsilon}$ only if almost all summands lie in a linear subspace whose translate is contained in the variety. Fourth, multi-copy decoupling inequalities that turn a quadratic anticoncentration event into $k$ simultaneous linear equations in independent random vectors, raising the exponent from $1/2$ toward $1$ as $k$ grows. Fifth, an optimal inverse linear Littlewood–Offord theorem, used to show the decoupled linear systems are unlikely to hit low-volume generalized arithmetic progressions.

What would settle it

For small $n$ (say up to 30), enumerate $3$-multilinear forms with coefficients in $\{-1,0,1\}$, retain only those whose every $4\times 4\times 4$ subtensor has partition rank at least 2, and compute the maximum value probability by exact enumeration over all sign configurations; exceeding $n^{-1+\varepsilon}$ for any fixed $\varepsilon>0$ would refute the flagship $n^{-1+\varepsilon}$ theorem.

Watch

Extended reading notes

Core claim

The paper's central claim is that algebraic structure, not coefficient sparsity, is the true obstruction to anticoncentration. For a $d$-multilinear form $f$ — a degree-$d$ polynomial whose variables split into $d$ types, one variable per type in each monomial — the coefficient tensor either can be made reducible (partition rank 1) by changing $\varepsilon n^d$ entries, or else $\sup_z \mathbb{P}[f(\xi_1,\dots,\xi_n)=z] \le n^{-1+\varepsilon}$; this is the repaired multilinear conjecture with the optimal $n^{-1}$ exponent. Over the complex numbers, a general quadratic polynomial that is not $\varepsilon n^2$-close to any reducible quadratic satisfies the power saving $n^{-13/24+\varepsilon}$, and if its quadratic part is not close to a form of rank less than $2k^2$, the bound improves to $n^{-1+2/k}$. The mechanism is local-to-global: sample small subtensors, classify the linear subspaces of the variety of reducible tensors, and reduce high-degree concentration to linear concentration via multi-copy decoupling and an inverse theorem for linear Littlewood–Offord. The original conjecture with $n^{-d/2}$ is false: the $d$-multilinear form $L_1\cdots L_d - L_{d+1}\cdots L_{2d}$ is robustly irreducible over $\mathbb{C}$ yet has $\mathbb{P}[f(\xi)=0]\ge \varepsilon/n$, so no bound better than roughly $1/n$ is possible in general.

Load-bearing premise

The argument leans on an imported quantitative description of when a signed sum of vectors concentrates on one value, and the hidden constants in that description are not tracked; if those constants were substantially worse than stated, the exponents $13/24$ and $1-2/k$ would weaken.

Editorial extensions

If this is right

  • A $d$-multilinear form has point probability at most $n^{-1+\varepsilon}$ unless it is $\varepsilon n^d$-close to reducible, settling the multilinear case at the best possible exponent up to $n^{\varepsilon}$.
  • Over $\mathbb{C}$, every robustly irreducible quadratic polynomial satisfies $\sup_z \mathbb{P}[f(\xi)=z] \le n^{-13/24+\varepsilon}$, a genuine power saving below the generic $n^{-1/2}$ bound.
  • For quadratic polynomials, guaranteeing that the quadratic part is not close to a form of rank less than $2k^2$ forces $\sup_z \mathbb{P}[f(\xi)=z] \le n^{-1+2/k}$; taking $k$ large approaches the $1/n$ barrier.
  • The example $f=L_1\cdots L_d - L_{d+1}\cdots L_{2d}$ is robustly irreducible even over $\mathbb{C}$ yet assigns probability at least $\varepsilon/n$ to $0$, so the original $n^{-d/2}$ conjecture fails for every $d\ge 3$.
  • The local-to-global lemmas yield constant-query property tests: a tensor can be certified close to partition rank 1 by sampling constantly many small subtensors, and a symmetric matrix close to rank $r$ by sampling small principal submatrices.

Reading between the lines

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

  • If the hidden constants in the imported inverse theorem were made explicit, the same induction could plausibly push the complex-quadratic exponent $13/24$ up toward $1$, matching the general conjecture that robust irreducibility alone yields $n^{-1+o(1)}$.
  • The counterexample family $P(L_1,\dots,L_k)$ links the Littlewood–Offord problem to counting integral points on varieties: a transfer of dimension-growth estimates into this discrete setting would be a route toward structural inverse theorems, an implication the authors flag but do not develop.
  • The multi-copy decoupling inequalities look portable: applying the same $k$-copy trick to degree-$d$ polynomials, not just quadratics, should raise the achievable exponent toward $n^{-d/2}$ whenever the associated tensor has high enough rank or strength.
  • The local-to-global lemmas suggest practical constant-time rank tests: sampling $O(1)$ random subtensors decides whether a huge tensor is close to partition rank 1, and the symmetric-matrix analogue gives a symmetry-preserving low-rank approximation algorithm.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies anticoncentration of degree-d polynomials of independent Rademacher variables, in the direction of the polynomial Littlewood–Offord problem. Its main results are: an optimal n^{-1+o(1)} point-probability bound for d-multilinear forms that are far from reducible (Theorem 1.9), a power-saving bound n^{-1/2-1/24+o(1)} for robustly irreducible complex quadratic polynomials (Theorem 1.10), and a rank-dependent bound n^{-1+2/k+o(1)} for quadratic polynomials whose quadratic part has high rank (Theorem 1.11). The paper also contains an appendix disproving Costello's original conjecture on multilinear forms, together with local-to-global lemmas for tensor reducibility and symmetric matrix rank that are of independent interest for property testing. The proofs are organized around explicit epsilon-delta parameters and are mostly self-contained, but the proof of the geometric Littlewood–Offord theorem in Appendix A contains a serious gap that affects the proof of Theorem 1.9.

Significance. If the results are correct, Theorem 1.9 essentially resolves the repaired multilinear version of Costello's conjecture at the optimal n^{-1} exponent, since examples from the multiplication-table problem show that no bound better than (log n)^O(1)/n is possible. Theorem 1.10 gives the first power-saving beyond the Meka–Nguyen–Vu bound under an algebraic irreducibility assumption in the complex quadratic case, and Theorem 1.11 provides a partial answer to the rank-based version of the problem. The local-to-global lemmas for tensors and symmetric matrices, and the appendix disproof of Costello's original conjecture, are substantial contributions in their own right. The paper is also commendable for stating all quantitative parameters explicitly rather than hiding them in unspecified constants. However, the central multilinear theorem relies on a flawed proof in Appendix A, so the paper is not yet in publishable form.

major comments (2)
  1. [Appendix A, proof of Theorem 2.5] The disjoint-bases claim in the proof of Theorem 2.5 is incorrect. The proof defines n_b = n - \epsilon n^{(d-b)/d}, lets b be minimal such that at least n_b of the vectors a_i lie in a common b-dimensional subspace W, and then asserts that minimality yields at least (n_b - n_{b-1})/b \ge \epsilon n/d^2 disjoint bases of W. For b=d this is false: n_d - n_{d-1} = \epsilon(n^{1/d}-1), so the number of disjoint bases is only about \epsilon n^{1/d}/d, not linear in n. A concrete example in d=2 is obtained by taking n-\sqrt{n} copies of (1,0) and \sqrt{n} copies of (0,1) with Z={0}; then no line contains n-\epsilon\sqrt{n} vectors, so b=2, yet only \sqrt{n} disjoint bases exist. Consequently the later choice m = \lfloor \epsilon n/d^2 \rfloor b is not legitimate, and the coefficient-counting argument yields only m^q = n^{q/d} nonzero coefficients rather than n^q. The resulting bound is n^{-1/(2d)+o(1)} instead of the claimed n^{-1/2+\epsilon}.
  2. [Section 6, proof of Theorem 6.1 via Theorem 2.4] The gap in Appendix A propagates to the proof of the paper's main multilinear theorem. Theorem 2.5 is used in the proof of Lemma 2.4 to obtain the n^{-1/2+\epsilon} bound in D2, and Lemma 2.4 is then an essential induction step in the proof of Theorem 6.1, which implies Theorem 1.9. Since the proof of Theorem 2.5 presented in the manuscript does not establish the claimed exponent, the proof of Theorem 1.9 is incomplete as written. The authors should either repair the disjoint-bases argument, or explicitly state that Theorem 2.5 is imported from the published paper [18] and remove the flawed self-contained proof.
minor comments (4)
  1. [Throughout] The text frequently switches between calling the same statement a Lemma and a Theorem (for example, Lemma 2.2, Lemma 2.4, and Lemma 2.7 are sometimes referred to as Theorems 2.2, 2.4, and 2.7). These labels should be made consistent.
  2. [Section 6] In the proof of Theorem 6.1, the event E_1 is defined using epsilon_1 = epsilon^3/32 and the bound |I'|^{-1/2+epsilon_1} is stated as <= n^{-1/2+epsilon/2}; since epsilon_1 is much smaller than epsilon/2 this is fine, but the display would be clearer if the inequality were written as |I'|^{-1/2+epsilon_1} <= n^{-1/2+epsilon/2} after using |I'| >= \epsilon n.
  3. [Appendix B, Claim B.5] After the proof of Claim B.5, the text says 'Theorem B.5 tells us...' but the statement is a claim, not a theorem; similarly, several equations in Section 10 refer to 'Theorem 10.2' and 'Theorem 10.5' where 'Lemma 10.2' and 'Lemma 10.5' would be more accurate. These numbering inconsistencies should be cleaned up.
  4. [Section 10.1] In the proof of Theorem 2.17, the exponent calculations in the estimates for the second and third terms are terse; for example, the step from n^{-3/2+5\delta} E[...] to n^{-2-1/6+7\delta} uses Theorem 10.6, but it would help the reader if the inequalities were displayed separately rather than compressed into one chain.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorems are derived from external imported results and from proofs contained in the paper; self-citations are not load-bearing.

full rationale

The central derivation chain for Theorem 1.9 is an induction on d. The base case d=2 is imported from Costello [9] (Theorem 6.2/6.3), an external theorem. The induction step uses Lemma 2.4, whose proof applies Theorem 2.5 and Theorem 2.6. Theorem 2.5 is stated as closely related to [18], but the paper gives a proof in Appendix A reducing it to the Meka-Nguyen-Vu polynomial Littlewood-Offord bound (Theorem 1.1), an external result; no step assumes the target theorem to prove the input. Theorems 1.10 and 1.11 similarly reduce to Theorems 2.13 and 2.17, whose proofs use the Nguyen-Vu inverse theorem (Theorem 2.10) and Halasz's theorem (Theorem 3.2), both external, together with decoupling lemmas proved in Section 8. All epsilons and deltas are explicitly quantified variables; there are no fitted constants, no output-dependent normalization, and no prediction that is defined in terms of the quantity being bounded. The self-citations ([18], [34], [35]) are either supported by an included proof or used for comparison and motivation; they are not the unique justification of a forced conclusion. The skeptical note about the claim in Appendix A concerning at least (n_b - n_{b-1})/b >= epsilon n/d^2 disjoint bases is a correctness or quantitative concern, not an identity between theorem and input; even if that estimate were wrong, it is a gap in a proof, not circularity by construction. No circular step is identified.

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

The central claims involve no fitted constants and no invented objects. All epsilons, deltas, and ranks are explicitly quantified variables in theorem statements, not parameters fitted to data. The paper depends on several deep published theorems, listed as axioms, but not on any self-referential normalization or on constants chosen after the fact.

assumptions (5)
  • standard math Meka-Nguyen-Vu polynomial Littlewood-Offord theorem (Theorem 1.1).
    Used as a black box in Appendix A and in the proof of Theorem 6.1 to convert coefficient-size lower bounds into point-probability upper bounds.
  • standard math Nguyen-Vu optimal inverse Littlewood-Offord theorem (Theorem 2.10).
    Basic tool in the proofs of Theorems 2.13 and 2.17; supplies the symmetric generalized arithmetic progression covering used for high-dimensional linear anticoncentration.
  • standard math Halasz's high-dimensional Littlewood-Offord theorem (Theorem 3.2) and its corollaries.
    Used throughout Sections 6, 9, and 10 to bound probabilities that random linear combinations lie in subspaces or have many zero coordinates.
  • standard math Costello's d=2 base theorem [9, Theorem 5] and Lemma 8.
    Base case of the induction for multilinear forms and the quantitative estimate behind Theorem 10.6(2); the paper extends the base via Theorem 6.3.
  • domain assumption Standard algebraic geometry facts over R and C: rank-1 matrices form a variety, linear subspaces are irreducible affine varieties, and a complex quadratic form is reducible iff its rank is at most 2.
    Used in the proof of Theorem 2.6 and in the proof of Theorem 1.10; these facts are standard and the paper explicitly restricts attention to F in {R,C}.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Algebraic aspects of the polynomial Littlewood-Offord problem." pith.science (2026). https://pith.science/paper/ZPVJOPEB

@misc{pith2026250523335,
  author       = {Pith},
  title        = {Pith review of: Algebraic aspects of the polynomial Littlewood-Offord problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZPVJOPEB}},
  note         = {Machine review of arXiv:2505.23335}
}
abstract

Consider a degree-$d$ polynomial $f(\xi_1,\dots,\xi_n)$ of independent Rademacher random variables $\xi_1,\dots,\xi_n$. To what extent can $f(\xi_1,\dots,\xi_n)$ concentrate on a single point? This is the so-called polynomial Littlewood-Offord problem. A nearly optimal bound was proved by Meka, Nguyen and Vu: the point probabilities are always at most about $1/\sqrt n$, unless $f$ is "close to the zero polynomial" (having only $o(n^d)$ nonzero coefficients). In this paper we prove several results supporting the general philosophy that the Meka-Nguyen-Vu bound can be significantly improved unless $f$ is "close to a polynomial with special algebraic structure", drawing some comparisons to phenomena in analytic number theory. In particular, one of our results is a corrected version of a conjecture of Costello on multilinear forms (in an appendix with Ashwin Sah and Mehtaab Sawhney, we disprove Costello's original conjecture).

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Geometric Littlewood-Offord problems via lattice point counting

    math.CO 2025-05 accept novelty 8.0 of 10

    Geometric Littlewood-Offord probabilities are bounded by counting lattice points, resolving conjectures for varieties, convex-position sets, and bounded Chow-rank polynomials.

Reference graph

Works this paper leans on

62 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [18]

    J. Fox, M. Kwan, and H. Spink,Geometric and o-minimal Littlewood–Offord problems, Ann. Probab.51(2023), no. 1, 101–126. 7, 8

    work page 2023

  2. [1]

    Alon and R

    N. Alon and R. Beigel,Lower bounds for approximations by low degree polynomials over z/sub m/, Proceedings 16th Annual IEEE Conference on Computational Complexity, CCC-01, IEEE Comput. Soc, 2001, p. 184–187. 36

    work page 2001

  3. [2]

    Balcan, Y

    M.-F. Balcan, Y. Li, D. P. Woodruff, and H. Zhang,Testing matrix rank, optimally, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2019, pp. 727–746. 6

    work page 2019

  4. [3]

    Bhattacharyya and Y

    A. Bhattacharyya and Y. Yoshida,Property testing—problems and techniques, Springer, Singapore, 2022. 6

    work page 2022

  5. [4]

    T. D. Browning and D. R. Heath-Brown,Counting rational points on hypersurfaces, J. Reine Angew. Math.584 (2005), 83–115. 4

    work page 2005

  6. [5]

    T. D. Browning, D. R. Heath-Brown, and P. Salberger,Counting rational points on algebraic varieties, Duke Math. J.132(2006), no. 3, 545–578. 5

    work page 2006

  7. [6]

    T. D. Browning,Quantitative arithmetic of projective varieties, Progress in Mathematics, vol. 277, Birkhäuser Verlag, Basel, 2009. 2

    work page 2009

  8. [7]

    Cohen and G

    A. Cohen and G. Moshkovitz,Partition and analytic rank are equivalent over large fields, Duke Math. J.172(2023), no. 12, 2433–2470. 5

    work page 2023

Show all 62 references
  1. [8]

    S. D. Cohen,The distribution of Galois groups and Hilbert’s irreducibility theorem, Proc. London Math. Soc. (3)43 (1981), no. 2, 227–250. 5

    1981

  2. [9]

    K. P. Costello,Bilinear and quadratic variants on the Littlewood-Offord problem, Israel J. Math.194(2013), no. 1, 359–394. 2, 3, 6, 12, 18, 33

    2013

  3. [10]

    K. P. Costello, T. Tao, and V. Vu,Random symmetric matrices are almost surely nonsingular, Duke Math. J.135 (2006), no. 2, 395–413. 2, 9, 11

    2006

  4. [11]

    K. P. Costello and V. H. Vu,The rank of random graphs, Random Structures Algorithms33(2008), no. 3, 269–285. 11

    2008

  5. [12]

    De la Pena and E

    V. De la Pena and E. Giné,Decoupling: from dependence to independence, Springer Science & Business Media, 1999. 9

    1999

  6. [13]

    de Zeeuw,A survey of Elekes-Rónyai-type problems, New trends in intuitive geometry, Bolyai Soc

    F. de Zeeuw,A survey of Elekes-Rónyai-type problems, New trends in intuitive geometry, Bolyai Soc. Math. Stud., vol. 27, János Bolyai Math. Soc., Budapest, 2018, pp. 95–124. 2

    2018

  7. [14]

    Doeblin,Sur les sommes d’un grand nombre de variables aléatoires indépendantes, Bull

    W. Doeblin,Sur les sommes d’un grand nombre de variables aléatoires indépendantes, Bull. Sci. Math63(1939), no. 2, 23–32. 2

    1939

  8. [15]

    Erdős,On a lemma of Littlewood and Offord, Bull

    P. Erdős,On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc.51(1945), 898–902. 2

    1945

  9. [16]

    Ferber, V

    A. Ferber, V. Jain, and Y. Zhao,On the number of Hadamard matrices via anti-concentration, Combin. Probab. Comput.31(2022), no. 3, 455–477. 5

    2022

  10. [17]

    Ford,The distribution of integers with a divisor in a given interval, Ann

    K. Ford,The distribution of integers with a divisor in a given interval, Ann. of Math. (2)168(2008), no. 2, 367–433. 5

    2008

  11. [19]

    Frieze and B

    A. Frieze and B. Pittel,Perfect matchings in random graphs with prescribed minimal degree, Mathematics and com- puter science. III, Trends Math., Birkhäuser, Basel, 2004, pp. 95–132. 14

    2004

  12. [20]

    Goldreich,Introduction to property testing, Cambridge University Press, Cambridge, 2017

    O. Goldreich,Introduction to property testing, Cambridge University Press, Cambridge, 2017. 6

    2017

  13. [21]

    W. T. Gowers and T. Karam,Equidistribution of high-rank polynomials with variables restricted to subsets ofFp, arXiv preprint arXiv:2209.04932 (2022). 4

    2022 arXiv

  14. [22]

    Green and T

    B. Green and T. Tao,The distribution of polynomials over finite fields, with applications to the Gowers norms, Contrib. Discrete Math.4(2009), no. 2, 1–36. 4

    2009

  15. [23]

    Halász,Estimates for the concentration function of combinatorial number theory and probability, Period

    G. Halász,Estimates for the concentration function of combinatorial number theory and probability, Period. Math. Hungar.8(1977), no. 3-4, 197–211. 5, 6, 10, 13

    1977

  16. [24]

    D. R. Heath-Brown,The density of rational points on curves and surfaces, Ann. of Math. (2)155(2002), no. 2, 553–595. 5

    2002

  17. [25]

    Howard,Estimates on the concentration function of sets inRd: Notes on lectures of Oskolkov,https://people

    R. Howard,Estimates on the concentration function of sets inRd: Notes on lectures of Oskolkov,https://people. math.sc.edu/howard/Notes/concentration.pdf, 2000. 5

    2000

  18. [26]

    Janzer,Polynomial bound for the partition rank vs the analytic rank of tensors, Discrete Anal

    O. Janzer,Polynomial bound for the partition rank vs the analytic rank of tensors, Discrete Anal. (2020), Paper No. 7, 18. 4, 5

    2020

  19. [27]

    D. M. Kane,The correct exponent for the Gotsman–Linial conjecture, Comput. Complexity23(2014), no. 2, 151–175. 2

    2014

  20. [28]

    Karam,High-rank subtensors of high-rank tensors, arXiv preprint arXiv:2207.08030 (2022)

    T. Karam,High-rank subtensors of high-rank tensors, arXiv preprint arXiv:2207.08030 (2022). 6

    2022 arXiv

  21. [29]

    Y. R. Katznelson,Singular matrices and a uniform bound for congruence groups ofSLn(Z), Duke Math. J.69(1993), no. 1, 121–136. 27

    1993

  22. [30]

    Y. R. Katznelson,Integral matrices of fixed rank, Proc. Amer. Math. Soc.120(1994), no. 3, 667–675. 27

    1994

  23. [31]

    D. J. Kleitman,On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors, Advances in Math.5(1970), 155–157 (1970). 2

    1970

  24. [32]

    Kolmogorov,Sur les propriétés des fonctions de concentrations de M

    A. Kolmogorov,Sur les propriétés des fonctions de concentrations de M. P. Lévy, Ann. Inst. H. Poincaré16(1958), 27–34. 2

    1958

  25. [33]

    Krauthgamer and O

    R. Krauthgamer and O. Sasson,Property testing of data dimensionality, Proceedings of the Fourteenth Annual ACM- SIAM Symposium on Discrete Algorithms (Baltimore, MD, 2003), ACM, New York, 2003, pp. 18–27. 6

    2003

  26. [34]

    Kwan and L

    M. Kwan and L. Sauermann,Resolution of the quadratic Littlewood–Offord problem, arXiv preprint arXiv:2312.13826. 2, 5, 11, 13

    arXiv

  27. [35]

    Kwan and L

    M. Kwan and L. Sauermann,An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs, Discrete Anal. (2020), Paper No. 12, 34. 4, 22

    2020

  28. [36]

    Lampert and T

    A. Lampert and T. Ziegler,Relative rank and regularization, Forum Math. Sigma12(2024), Paper No. e29, 26. 5 34

    2024

  29. [37]

    Y. Li, Z. Wang, and D. P. Woodruff,Improved testing of low rank matrices, Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, KDD ’14, ACM, August 2014, p. 691–700. 6

    2014

  30. [38]

    J. E. Littlewood and A. C. Offord,On the number of real roots of a random algebraic equation. III, Rec. Math. [Mat. Sbornik] N.S.12(54)(1943), 277–286. 2

    1943

  31. [39]

    Lovett,The analytic rank of tensors and its applications, Discrete Anal

    S. Lovett,The analytic rank of tensors and its applications, Discrete Anal. (2019), Paper No. 7, 10. 5

    2019

  32. [40]

    12(2016), Paper No

    R.Meka, O.Nguyen, andV.Vu,Anti-concentration for polynomials of independent random variables, TheoryComput. 12(2016), Paper No. 11, 16. 1, 2

    2016

  33. [41]

    Milićević,Polynomial bound for partition rank in terms of analytic rank, Geom

    L. Milićević,Polynomial bound for partition rank in terms of analytic rank, Geom. Funct. Anal.29(2019), no. 5, 1503–1530. 4, 5

    2019

  34. [42]

    Moshkovitz and D

    G. Moshkovitz and D. G. Zhu,Quasi-linear relation between partition and analytic rank, arXiv preprint arXiv:2211.05780 (2022). 5

    2022 arXiv

  35. [43]

    Naslund,Exponential bounds for the Erdős-Ginzburg-Ziv constant, J

    E. Naslund,Exponential bounds for the Erdős-Ginzburg-Ziv constant, J. Combin. Theory Ser. A174(2020), 105185,

    2020

  36. [44]

    Naslund,The partition rank of a tensor andk-right corners inFn q, J

    E. Naslund,The partition rank of a tensor andk-right corners inFn q, J. Combin. Theory Ser. A174(2020), 105190,

    2020

  37. [45]

    Nguyen and V

    H. Nguyen and V. Vu,Optimal inverse Littlewood–Offord theorems, Adv. Math.226(2011), no. 6, 5298–5319. 6, 10

    2011

  38. [46]

    H. H. Nguyen,Inverse Littlewood–Offord problems and the singularity of random symmetric matrices, Duke Math. J. 161(2012), no. 4, 545–586. 6

    2012

  39. [47]

    H. H. Nguyen and V. H. Vu,Small ball probability, inverse theorems, and applications, Erdős centennial, Bolyai Soc. Math. Stud., vol. 25, János Bolyai Math. Soc., Budapest, 2013, pp. 409–463. 2

    2013

  40. [48]

    Pila,Density of integral and rational points on varieties, no

    J. Pila,Density of integral and rational points on varieties, no. 228, 1995, Columbia University Number Theory Seminar (New York, 1992), pp. 4, 183–187. 5

    1995

  41. [49]

    Razborov and E

    A. Razborov and E. Viola,Real advantage, ACM Trans. Comput. Theory5(2013), no. 4, Art. 17, 8. 2

    2013

  42. [50]

    Ron,Property testing: A learning theory perspective, now Publishers Inc, 2007

    D. Ron,Property testing: A learning theory perspective, now Publishers Inc, 2007. 6

    2007

  43. [51]

    Rosiński and G

    J. Rosiński and G. Samorodnitsky,Symmetrization and concentration inequalities for multilinear forms with applica- tions to zero-one laws for Lévy chaos, Ann. Probab.24(1996), no. 1, 422–437. 2, 9, 11

    1996

  44. [52]

    Rudelson and R

    M. Rudelson and R. Vershynin,The Littlewood–Offord problem and invertibility of random matrices, Adv. Math.218 (2008), no. 2, 600–633. 6

    2008

  45. [53]

    Salberger,Counting rational points on projective varieties, Proc

    P. Salberger,Counting rational points on projective varieties, Proc. Lond. Math. Soc. (3)126(2023), no. 4, 1092–1133. 5

    2023

  46. [54]

    slice rank

    W. Sawin and T. Tao,Notes on the “slice rank” of tensors,https://terrytao.wordpress.com/2016/08/24/ notes-on-the-slice-rank-of-tensors/., 2011. 8

    2016

  47. [55]

    W. M. Schmidt,The density of integer points on homogeneous varieties, Acta Math.154(1985), no. 3-4, 243–296. 5

    1985

  48. [56]

    Sidorenko,A correlation inequality for bipartite graphs, Graphs Combin.9(1993), no

    A. Sidorenko,A correlation inequality for bipartite graphs, Graphs Combin.9(1993), no. 2, 201–204. 11

    1993

  49. [57]

    Tao,A symmetric formulation of the croot-lev-pach-ellenberg-gijswijt capset bound,https://terrytao.wordpress

    T. Tao,A symmetric formulation of the croot-lev-pach-ellenberg-gijswijt capset bound,https://terrytao.wordpress. com/2016/05/18/a-symmetric-formulation-of-the-croot-lev-pach-ellenberg-gijswijt-capset-bound/., 2011. 8

    2016

  50. [58]

    Tao and V

    T. Tao and V. Vu,A sharp inverse Littlewood–Offord theorem, Random Structures Algorithms37(2010), no. 4, 525–539. 6

    2010

  51. [59]

    Tao and V

    T. Tao and V. H. Vu,Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math. (2)169(2009), no. 2, 595–632. 6

    2009

  52. [60]

    Vermeulen,Dimension growth for affine varieties, Int

    F. Vermeulen,Dimension growth for affine varieties, Int. Math. Res. Not. IMRN (2024), no. 15, 11464–11483. 5

    2024

  53. [61]

    Vu,Anti-concentration inequalities for polynomials, A journey through discrete mathematics, Springer, Cham, 2017, pp

    V. Vu,Anti-concentration inequalities for polynomials, A journey through discrete mathematics, Springer, Cham, 2017, pp. 801–810. 2

    2017

  54. [62]

    approximate factorisation

    M. N. Walsh,Bounded rational points on curves, Int. Math. Res. Not. IMRN (2015), no. 14, 5644–5658. 5 AppendixA.A Littlewood–Offord theorem for varieties In this appendix we prove Theorem 2.5, giving a bound of the formP[ξ1⃗ a1 +· · ·+ξn⃗ an ∈ Z]≤ n−1/2+o(1) unless almost all ...

    2015

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.