Let B be a set of complex numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = bb’ | b,b’ ∈ B cannot be greater than O((nlog²n)/(loglogn)) and present an example of a product set containing an arithmetic progression of length Ω(nlogn). For sets of complex numbers we obtain the upper bound .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-1,
author = {Dmitrii Zhelezov},
title = {Product sets cannot contain long arithmetic progressions},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {299-307},
zbl = {1318.11012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-1}
}
Dmitrii Zhelezov. Product sets cannot contain long arithmetic progressions. Acta Arithmetica, Tome 166 (2014) pp. 299-307. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-4-1/