A Quantitative Bound For Szemer\'edi's Theorem for a Complexity One Polynomial Progression over $\mathbb{Z}/N\mathbb{Z}$

TL;DR

This paper establishes a quantitative upper bound on the size of subsets of rac{N}{ ext{log}^{(O(1))}(N)} ext{N} where certain polynomial progressions are absent, extending Szemere9di's theorem to complexity one polynomial progressions over finite cyclic groups.

Contribution

It adapts Peluse's degree lowering technique to quadratic Fourier analysis to provide explicit bounds for polynomial progressions of complexity one, advancing understanding of polynomial Szemere9di theorems.

Findings

Sets lacking the progression are of size at most rac{N}{ ext{log}^{(O(1))}(N)}

Method extends to polynomial progressions with polynomial-type bounds on true complexity

Provides quantitative bounds for polynomial Szemere9di-type theorems

Abstract

Let $N$ be a large prime and $P, Q \in \mathbb{Z}[x]$ two linearly independent polynomials with $P(0) = Q(0) = 0$. We show that if a subset $A$ of $\mathbb{Z}/N\mathbb{Z}$ lacks a progression of the form $(x, x + P(y), x + Q(y), x + P(y) + Q(y))$, then $$|A| \le O\left(\frac{N}{\log_{(O(1))}(N)}\right)$$ where $\log_{C}(N)$ is an iterated logarithm of order $C$ (e.g., $\log_{2}(N) = \log\log(N)$). To establish this bound, we adapt Peluse's (2018) degree lowering argument to the quadratic Fourier analysis setting to obtain quantitative bounds on the true complexity of the above progression. Our method also shows that for a large class of polynomial progressions, if one can establish polynomial-type bounds on the true complexity of those progressions, then one can establish polynomial-type bounds on Szemer\'edi's theorem for that type of polynomial progression.