A blurred view of Van der Waerden type theorems

TL;DR

This paper explores approximate versions of classical theorems on arithmetic progressions, analyzing how near-approximations affect the existence and properties of such progressions in various settings.

Contribution

It extends Van der Waerden, Szemeredi, and Furstenberg-Katznelson theorems to approximate arithmetic progressions, providing new insights into their numerical and structural aspects.

Findings

Approximate arithmetic progressions exist under certain conditions.

Numerical bounds for epsilon-approximations are established.

Extensions to higher dimensions are analyzed.

Abstract

Let $AP_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\epsilon>0$ we call a set $AP_k(\epsilon)=\{x_0,\ldots,x_{k-1}\}$ an $\epsilon$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<\epsilon d$ holds for all $i\in\{0,1\ldots,k-1\}$. Complementing earlier results of Dumitrescu, in this paper we study numerical aspects of Van der Waerden, Szemeredi and Furstenberg-Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their $\epsilon$-approximation.