A blurred view of Van der Waerden type theorems

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.