Product sets cannot contain long arithmetic progressions

Dmitrii Zhelezov. "Product sets cannot contain long arithmetic progressions." Acta Arithmetica 163.4 (2014): 299-307. <http://eudml.org/doc/279420>.

@article{DmitriiZhelezov2014,
abstract = {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 $O(n^\{3/2\})$.},
author = {Dmitrii Zhelezov},
journal = {Acta Arithmetica},
keywords = {product set; arithmetic progression; convex sequence; complex number},
language = {eng},
number = {4},
pages = {299-307},
title = {Product sets cannot contain long arithmetic progressions},
url = {http://eudml.org/doc/279420},
volume = {163},
year = {2014},
}

TY - JOUR
AU - Dmitrii Zhelezov
TI - Product sets cannot contain long arithmetic progressions
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 4
SP - 299
EP - 307
AB - 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 $O(n^{3/2})$.
LA - eng
KW - product set; arithmetic progression; convex sequence; complex number
UR - http://eudml.org/doc/279420
ER -