[Paper Review] Grünwald version of van der Waerden's theorem for semi-modules

Xiongping Dai|arXiv (Cornell University)|Dec 29, 2015

Limits and Structures in Graph Theory25 references3 citations

TL;DR

This paper generalizes Grünwald's version of van der Waerden's theorem to semi-modules over *-semirings by establishing a Regional Multiple Recurrence Theorem, proving that in any finite coloring of a semi-module, one color class contains syndetic dilations of any finite set. The key result shows that each uniformly almost periodic point is multiply almost periodic, extending classical partition regularity to algebraic structures beyond integers.

ABSTRACT

Let $(G,\pmb{+})$ be any given semimodule over a discrete semiring $(R,+,\cdot)$ with a finite coloring, say $G=B_1\cup\dotsm\cup B_q$. By establishing a Regional Multiple Recurrence Theorem for semimodules, we prove that one of the colors $j$ has the property that if $F\subseteq G$ is any finite set, then one can find some "syndetic" subset $D_F$ of $(R,+)$ such that for each $d\in D_F$ there is some $a\in B_j$ with $a\pmb{+}dF\subseteq B_j$. This in turn implies that each Bohr almost periodic point is multiply uniformly recurrent.

Motivation & Objective

• To extend Grünwald’s version of van der Waerden’s theorem from $\mathbb{N}^m$ to semi-modules over semi-rings.
• To establish a Regional Multiple Recurrence Theorem for semi-modules as a foundational tool.
• To prove that in any finite coloring of a semi-module, one color class contains syndetic dilations of any finite set $F$.
• To show that uniformly almost periodic points in such systems are multiply almost periodic.
• To address the open Schur-Brauer-type conjecture on homothetic copies in semi-modules.

Proposed method

• Formalizing semi-modules over *-semirings with scalar multiplication and additive structure.
• Defining syndetic subsets in the semi-ring $R$ via finite sets $K$ such that $K + t \cap S \neq \emptyset$ for all $t \in R$.
• Using ultrafilter methods and Stone-Čech compactification to analyze recurrence in semi-modules.
• Proving a finitary version of the main result via limit points of colorings on increasing chains $M_n \subset M$.
• Applying the Regional Multiple Recurrence Theorem to show existence of $a \in M$, $d \in R \setminus \{0\}$ such that $a + dF \subseteq B_j$ for some color $j$.
• Establishing that $R$ being a *-semiring ensures nontrivial syndetic sets contain nonzero elements, avoiding trivial solutions.

Experimental results

Research questions

• RQ1 Does every finite coloring of a semi-module over a *-semiring contain a syndetic dilation of any finite set $F$ in one color class?
• RQ2 Can the classical Grünwald-van der Waerden theorem be extended to semi-modules over arbitrary semi-rings?
• RQ3 Is every uniformly almost periodic point in a semi-module system also multiply almost periodic?
• RQ4 What conditions on the semi-ring $R$ ensure that syndetic subsets contain non-zero elements?
• RQ5 Does the Schur-Brauer conjecture hold for semi-modules, i.e., does $a + Fb \subseteq B_j$ hold for some $b \neq \mathbf{o}$ and $a \in M$?

Key findings

• For any finite coloring $M = B_1 \cup \dotsm \cup B_q$ of a semi-module $M$ over a *-semiring $R$, one color class $B_j$ contains a syndetic dilation $a + dF$ for any finite $F \subset M$ and $d \in R \setminus \{0\}$.
• The Regional Multiple Recurrence Theorem guarantees the existence of such syndetic sets $D_F \subset R$ such that for each $d \in D_F$, there exists $a \in B_j$ with $a + dF \subseteq B_j$.
• The result implies that every uniformly almost periodic point in the system is multiply almost periodic, extending Furstenberg’s recurrence results.
• The finitary version (Theorem 3.25) shows that for any $n$ large enough, any $q$-coloring of $M_n$ contains a homothetic copy $a + dF$ in one color, with $d \neq 0$.
• The *-semiring condition ensures that syndetic sets contain non-zero elements, which is essential to avoid trivial solutions with $d = 0$.
• The paper confirms the conjecture that $a + Fb \subseteq B_j$ for some $b \neq \mathbf{o}$ and $a \in M$ under the Schur-Brauer-type condition, generalizing classical results.