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A blurred view of Van der Waerden type theorems
Published online by Cambridge University Press: 26 November 2021
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.
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- © The Author(s), 2021. Published by Cambridge University Press
Footnotes
Dedicated to the memory of Ronald Graham
The first author was supported by NSF grant DMS 1764385.
The second author was partially supported by NSF grant DMS 1764385.