An algebraic inverse theorem for the quadratic Littlewoodâ Offord problem

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.Non UBCUnreviewedAuthor affiliation: Stanford UniversityPostdoctora