Abstract
The classical Erdős–Littlewood–Offord theorem says that for nonzero vectors , any , and uniformly random , we have . In this paper, we show that whenever S is definable with respect to an o-minimal structure (e.g., this holds when S is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.
Funding Statement
The first author was supported by a Packard Fellowship and by NSF Award DMS-1855635. The second author was supported by NSF Award DMS-1953990.
Acknowledgements
The authors would like to thank Jonathan Pila for a number of insightful comments and suggestions.
Citation
Jacob Fox. Matthew Kwan. Hunter Spink. "Geometric and o-minimal Littlewood–Offord problems." Ann. Probab. 51 (1) 101 - 126, January 2023. https://doi.org/10.1214/22-AOP1590
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