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An algebraic inverse theorem for the quadratic Littlewoodâ Offord problem

Banff International Research Station for Mathematical Innovation and Discovery

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the ``inverse'' question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this talk we present some results of an algebraic flavour, showing that if $f$ has point probabilities much larger than $1/n$ then it must be close to a quadratic form with low rank. We also give an application to Ramsey graphs, asymptotically answering a question of Kwan, Sudakov and Tran. Joint work with Lisa Sauermann.

Mathematics; Combinatorics; Probability theory and stochastic processes

video/mp4

Author affiliation: Stanford University

2020-09-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75863

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/