Approximate quadratic varieties
Luka Mili\'cevi\'c

TL;DR
This paper extends classical additive combinatorics results to approximate quadratic structures, showing that sets with many additive cubes and certain linear restrictions closely resemble exact quadratic varieties.
Contribution
It introduces a new notion of approximate quadratic varieties and proves a structure theorem linking these sets to exact quadratic varieties.
Findings
Sets with many additive cubes and linear restrictions are close to exact quadratic varieties.
The paper defines a new class of approximate quadratic varieties with specific density and uniformity conditions.
Main result: such approximate quadratic varieties have large intersections with true quadratic varieties.
Abstract
A classical result in additive combinatorics, which is a combination of Balog-Szemer\'edi-Gowers theorem and a variant of Freiman's theorem due to Ruzsa, says that if a subset of contains at least additive quadruples, then there exists a subspace , comparable in size to , such that . Motivated by the fact that higher order approximate algebraic structures play an important role in the theory of uniformity norms, it would be of interest to find higher order analogues of the mentioned result. In this paper, we study a quadratic version of the approximate property in question, namely what it means for a set to be an approximate quadratic variety. It turns out that information on the number of additive cubes, which are 8-tuples of the form , in a set…
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Taxonomy
TopicsMathematical Approximation and Integration · Limits and Structures in Graph Theory · Polynomial and algebraic computation
