For every fixed integer d≥ 1, there exists a constant C_d>0 such that the following holds. If F:R^n→R is a degree-d multilinear polynomial containing r degree-d monomials whose sets of variables are pairwise disjoint, and xi_1,…,xi_n are independent Rademacher random variables, then Pbigl[F(xi_1,…,xi_n)=0 bigr]≤ C_d r^-1/2; equivalently, Pbigl[F(xi_1,…,xi_n)=0 bigr]=O_d(r^-1/2).

Polynomial Littlewood–Offord conjecture

The conjecture seeks the optimal anti-concentration bound for random evaluations of polynomials. The general case was previously known only with a logarithmic or subpolynomial loss, while the linear and quadratic cases were understood more completely.

Known results

The optimal-order bound is known for degrees $d=1$ and $d=2$ ; general $d$ previously incurred a factor $(\log b)^{O_d(1)}$ (Kane; Meka, Nguyen, and Vu).
Kwan, Sah, and Sawhney disproved Costello's original stronger conjecture for $d\ge 3$ .
The repaired conjecture is proved for $d$ -multilinear forms, and partial bounds are known for complex quadratics and high-rank quadratic parts.
A quadratic robust-dependence theorem gives an upper bound of order $m^{-1/2}$ under its stated hypothesis.

October 2026 claimed resolution

On October 6, 2026, Alexandr Grebennikov's manuscript Optimal bound for the polynomial Littlewood-Offord problem claimed removal of the polylogarithmic factor for polynomials containing $r$ disjoint degree- $d$ monomials, which would settle the stated conjecture and a related total-influence conjecture. The manuscript credits GPT-6 Pro, but the claim is unrefereed and has no independent mathematical corroboration in the retrieved evidence.

Current status (as of October 2026): The general conjecture remains unverified; the latest manuscript claims a complete proof, while established results cover important special cases and prior bounds with losses.

Polynomial Littlewood–Offord conjecture

Primary source

Additional references

Progress summary

Solutions 0