On the Polynomial Szemerédi Theorem in Finite Commutative Rings

TL;DR

This work proves a polynomial Szemerédi-type theorem for independent families of multivariable integer polynomials over general finite commutative rings with large torsion, establishing quantitative zero-density and universal density-type consequences. The authors develop and combine a suite of novel tools tailored to rings, including Z_N-height, essential distinctness modulo N, and a ring-adapted PET induction, supplemented by Fourier-analytic bounds on multicharacter sums and a Gowers-norm framework. The main theorem shows that, for rings with lpf(char(R)) sufficiently large, the average over x and y of f_0(x)f_1(x+P_1(y))…f_m(x+P_m(y)) concentrates near the expected product of averages, with an error decaying in the ring’s torsion via gamma. From this, they derive zero-density and other combinatorial consequences, including robust counting of nondegenerate configurations in large-subset regimes across broad families of rings. The results generalize prior finite-field cases and illuminate how asymptotic total ergodicity-like phenomena extend to rings with zero divisors, providing both structural and quantitative insights into polynomial configurations in algebraic combinatorics.

Abstract

The polynomial Szemerédi theorem implies that, for any $δ\in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of $\{1,\ldots, N\}$ of cardinality at least $δN$ contains a nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When the polynomials are assumed independent, one can expect a sharper result to hold over finite fields, special cases of which were proven recently, culminating with arXiv:1802.02200, which deals with the general case of independent polynomials. One goal of this article is to explain these theorems as the result of joint ergodicity in the presence of asymptotic total ergodicity. Guided by this concept, we establish, over general finite commutative rings, a version of the polynomial Szemerédi theorem for independent polynomials $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y_1,\ldots, y_n]$, deriving new combinatorial consequences, such as the following. Let $\mathcal R$ be a collection of finite commutative rings subject to a mild condition on their torsion. There exists $γ\in (0,1)$ such that, for every $R \in \mathcal R$, every subset $A \subset R$ of cardinality at least $|R|^{1-γ}$ contains a nontrivial configuration $\{x,x+P_1(y),\ldots, x+P_m(y)\}$ for some $(x,y) \in R \times R^n$, and, moreover, for any subsets $A_0,\ldots, A_m \subset R$ such that $|A_0|\cdots |A_m| \geq |R|^{(m+1)(1-γ)}$, there is a nontrivial configuration $(x, x+P_1(y), \ldots, x+P_m(y)) \in A_0\times \cdots \times A_m$. The fact that general rings have zero divisors is the source of many obstacles, which we overcome; for example, by studying character sums, we develop a bound on the number of roots of an integer polynomial over a general finite commutative ring, a result which is of independent interest.

Theorems & Definitions (124)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• proof
• Theorem 1.4
• Theorem 1.7
• Proposition 1.8
• Remark 1.9
• Corollary 1.10
• proof