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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{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 -