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Algebraic aspects of the polynomial Littlewood-Offord problem

Zhihan Jin, Matthew Kwan, Lisa Sauermann, Yiting Wang

TL;DR

The paper advances the algebraic understanding of the polynomial Littlewood--Offord problem by showing that unless a degree-$d$ polynomial $f$ is near a structured low-complexity form, one can improve anticoncentration beyond the general $n^{-1/2+o(1)}$ bound. It introduces a robust framework combining local-to-global tensor and matrix rank principles, decoupling, and inverse Littlewood--Offord theorems to establish power-saving anticoncentration results for multilinear and quadratic polynomials; in particular, optimal results are obtained for $d$-multilinear forms (with a $1$-exponent stability) and sharp bounds in the complex quadratic case. The work also disproves Costello’s conjecture in general via a counterexample and connects the polynomial LO problem to analytic number theory through affine-point-density considerations, suggesting deeper arithmetic underpinnings. Methodologically, the paper develops tensor- and matrix-rank property testing, refines decoupling techniques, and uses inverse-LO theorems to bridge high-dimensional random sums with low-dimensional algebraic structure, achieving new inverse-type results for higher-degree polynomials and clarifying the role of robust irreducibility in anticoncentration.

Abstract

Consider a degree-$d$ polynomial $f(ξ_1,\dots,ξ_n)$ of independent Rademacher random variables $ξ_1,\dots,ξ_n$. To what extent can $f(ξ_1,\dots,ξ_n)$ 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).

Algebraic aspects of the polynomial Littlewood-Offord problem

TL;DR

The paper advances the algebraic understanding of the polynomial Littlewood--Offord problem by showing that unless a degree- polynomial is near a structured low-complexity form, one can improve anticoncentration beyond the general bound. It introduces a robust framework combining local-to-global tensor and matrix rank principles, decoupling, and inverse Littlewood--Offord theorems to establish power-saving anticoncentration results for multilinear and quadratic polynomials; in particular, optimal results are obtained for -multilinear forms (with a -exponent stability) and sharp bounds in the complex quadratic case. The work also disproves Costello’s conjecture in general via a counterexample and connects the polynomial LO problem to analytic number theory through affine-point-density considerations, suggesting deeper arithmetic underpinnings. Methodologically, the paper develops tensor- and matrix-rank property testing, refines decoupling techniques, and uses inverse-LO theorems to bridge high-dimensional random sums with low-dimensional algebraic structure, achieving new inverse-type results for higher-degree polynomials and clarifying the role of robust irreducibility in anticoncentration.

Abstract

Consider a degree- polynomial of independent Rademacher random variables . To what extent can 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 , unless is "close to the zero polynomial" (having only nonzero coefficients). In this paper we prove several results supporting the general philosophy that the Meka-Nguyen-Vu bound can be significantly improved unless 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).

Paper Structure

This paper contains 19 sections, 31 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Further directions, and speculations
  3. Connections to analytic number theory
  4. Further questions
  5. Inverse theorems
  6. Property testing for (tensor) rank
  7. Organisation of the paper
  8. Notation
  9. Proof outlines
  10. $d$-multilinear forms
  11. Quadratic polynomials
  12. Bounds in terms of the rank
  13. A power-saving improvement in the "robustly irreducible" case
  14. Preliminaries
  15. A local-to-global lemma for reducibility of tensors
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix $d\ge 1$ and $\varepsilon>0$, and let $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$ be the field of real or complex numbers. Let $n$ be sufficiently large (in terms of $\varepsilon,d$). Let $f\in\mathbb{F}[x_{1},\dots,x_{n}]$ be an $n$-variable polynomial of degree $d$. Then, at least one of the foll

Theorems & Definitions (73)

  • Theorem 1.1
  • Remark 1.2
  • Conjecture 1.3
  • Proposition 1.4
  • Remark 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.11
  • ...and 63 more