Gr\"unwald version of van der Waerden's theorem for semi-modules

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.