Overview
This paper investigates the characteristics of sets of large returns, specifically those defined by polynomial multi-correlation functions. These sets, denoted $R_\epsilon^{p_1,...,p_L}(A)$, are central to ergodic theoretical proofs of the polynomial and IP-polynomial Szemeredi theorems, as well as implications derived from the Density Polynomial Hales-Jewett conjecture (DPHJ). The research establishes conditions under which these sets exhibit syndeticity and the A-IP$^*$ property, which is a stronger form of syndeticity. Additionally, the study presents new combinatorial results related to such sets.
Research Context
The study of discrete sets of large returns is motivated by their role in ergodic theoretical frameworks for fundamental theorems in combinatorics. The definition of $R_\epsilon^{p_1,...,p_L}(A)$ involves $L$ non-constant polynomials $p_1,...,p_L \in \mathbb{Z}[x_1,...,x_d]$ with zero constant terms. The sets are defined as:
$$ R_\epsilon^{p_1,...,p_L}(A):=\{n\in \mathbb{Z}^d\,|\,\mu(A\cap T_1^{-p_1( n)}A\cap\cdots\cap T_L^{-p_L(n)}A) \ge \mu^{L+1}(A)-\epsilon\} $$Here, $T_j$ are commuting, invertible $\mu$-preserving transformations, $A$ is a measurable set, and $\epsilon > 0$. The broader context includes the Density Polynomial Hales-Jewett conjecture and its ergodic-theoretical and combinatorial consequences.
Findings
- A set of the form $R_\epsilon^{p_1,...,p_L}(A)$ is syndetic if and only if the polynomials $p_1,...,p_L$ are linearly independent. This finding addresses a question posed by Frantzikinakis-Kuca.
- The linear independence of $p_1,...,p_L$ implies that every set of the form $R_\epsilon^{p_1,...,p_L}(A)$ possesses the A-IP$^*$ property. This property is described as an "almost" IP$^*$ property and is stronger than syndeticity.
- A new combinatorial result is established: If $p_1,...,p_L$ are linearly independent, for any set $E\subseteq \mathbb{Z}^D$ with upper Banach density $d^*(E) > 0$, any non-zero vectors $v_1,..., v_L\in \mathbb{Z}^D$, and any $\epsilon > 0$, the set $$ S_\epsilon^{p_1,...,p_L}(E):=\{ n\in \mathbb{Z}^d\,|\,d^*(E\cap (E-p_1(n)v_1)\cap \cdots\cap (E-p_L(n)v_L)) \ge (d^*(E))^{L+1}-\epsilon\} $$ is A-IP$^*$.
- It is further demonstrated that when $D=1$, the combinatorial result described above is sharp, meaning the A-IP$^*$ property cannot be upgraded to IP$^*$.
- The techniques developed in this paper indicate that an amplified form of the IP-polynomial Szemeredi theorem, conjectured by Bergelson-McCutcheon, follows from the Density Polynomial Hales-Jewett conjecture.
Why This Matters
The results directly contribute to understanding the structural properties of sets central to ergodic theory and combinatorics. By clarifying the conditions under which these sets exhibit properties like syndeticity and A-IP$^*$, the research provides foundational insights for further development in these areas.