Inverse Littlewood–Offord meta-conjecture

Characterize the arithmetic structure forced on a vector v when its Littlewood–Offord concentration function ρ₀(v) is large.

Background

The paper presents inverse Littlewood–Offord theory as the study of structural consequences of unusually large concentration probabilities. It states a broad meta-conjecture asserting that large concentration must arise from arithmetic structure, while noting that existing structural results cover only certain probability ranges.

References

They suggested the following ``meta-conjecture'' that has guided much subsequent research.

\begin{center} If $\rho_0(v)$ is ``large'' then $v$ must exhibit arithmetic structure. \end{center}

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 4, subsection “Littlewood--Offord theory,” subsection “Inverse Littlewood Offord theory”