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An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs
Discrete analysis (2020) Cet article a éte moissonné depuis la source Scholastica

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Consider a quadratic polynomial $f\left(ξ_{1},\dots,ξ_{n}\right)$ 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 paper we prove 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.
Publié le : 
@article{DAS_2020_a8,
     author = {Matthew Kwan and Lisa Sauermann},
     title = {An algebraic inverse theorem for the quadratic {Littlewood-Offord} problem, and an application to {Ramsey} graphs},
     journal = {Discrete analysis},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2020_a8/}
}
TY  - JOUR
AU  - Matthew Kwan
AU  - Lisa Sauermann
TI  - An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs
JO  - Discrete analysis
PY  - 2020
UR  - http://geodesic.mathdoc.fr/item/DAS_2020_a8/
LA  - en
ID  - DAS_2020_a8
ER  - 
%0 Journal Article
%A Matthew Kwan
%A Lisa Sauermann
%T An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs
%J Discrete analysis
%D 2020
%U http://geodesic.mathdoc.fr/item/DAS_2020_a8/
%G en
%F DAS_2020_a8
Matthew Kwan; Lisa Sauermann. An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs. Discrete analysis (2020). http://geodesic.mathdoc.fr/item/DAS_2020_a8/