Variants of Raimi's theorem

Abstract: There exists $E\subseteq \mathbb{N}$ such that, whenever $r\in \mathbb{N}$ and $\mathbb{N}=\bigcup_{i=1}^rD_i$ there exist $i\in\{1,2,\ldots,r\}$ and $k\in \mathbb{N}$ such that $(D_i+k)\cap E$ is infinite and $(D_i+k)\setminus E$ is infinite.

A new proof of the theorem is due to N. Hindman, then to Bergelson and Weiss, and the generalization to the author. In the present talk, we give an outline of the new proofs and the generalization and some variations are discussed in different structures (e.g. in $\Z_n^k$, in $SL_2(\mathbb F_p)$.)

These variations are joint work with J\'anos Pach and Thang Pham.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)