A refinement of Cauchy-Schwarz complexity
Pablo Candela, Diego Gonz\'alez-S\'anchez, Bal\'azs Szegedy

TL;DR
This paper introduces a refined complexity measure for systems of linear forms, called sequential Cauchy-Schwarz complexity, which provides sharper bounds for controlling averages of functions and improves understanding of polynomial true-complexity in additive combinatorics.
Contribution
The paper defines the sequential Cauchy-Schwarz complexity, refines existing notions, and applies it to systems over finite fields, yielding polynomial bounds and new proofs of inverse theorems.
Findings
Sequential Cauchy-Schwarz complexity is at most (k, l) for certain systems, providing tighter control.
Polynomial true-complexity bounds are established for systems like arithmetic progressions.
The approach leads to new proofs of inverse theorems for Gowers norms on finite fields.
Abstract
We introduce a notion of complexity for systems of linear forms, called sequential Cauchy-Schwarz complexity, which is parametrized by two positive integers and refines the notion of Cauchy-Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy-Schwarz complexity at most then any average of 1-bounded functions over this system is controlled by the -th power of the Gowers -norms of the functions. For this agrees with Cauchy-Schwarz complexity, but for there are systems that have sequential Cauchy-Schwarz complexity at most whereas their Cauchy-Schwarz complexity is greater than . Our main application illustrates this with systems over a prime field that are denoted by and can be viewed as -dimensional arithmetic progressions of…
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Taxonomy
Topicssemigroups and automata theory · Commutative Algebra and Its Applications · Advanced Algebra and Logic
