Abstract
Abstract
Let
$\mathrm{AP}_k=\{a,a+d,\ldots,a+(k-1)d\}$
be an arithmetic progression. For
$\varepsilon>0$
we call a set
$\mathrm{AP}_k(\varepsilon)=\{x_0,\ldots,x_{k-1}\}$
an
$\varepsilon$
-approximate arithmetic progression if for some a and d,
$|x_i-(a+id)|<\varepsilon d$
holds for all
$i\in\{0,1\ldots,k-1\}$
. Complementing earlier results of Dumitrescu (2011, J. Comput. Geom.2(1) 16–29), in this paper we study numerical aspects of Van der Waerden, Szemerédi and Furstenberg–Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their
$\varepsilon$
-approximation.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science