Geodesic
Product sets cannot contain long arithmetic progressions
Dmitrii Zhelezov1
1 Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg 41296 Gothenburg, Sweden
Acta Arithmetica, Tome 163 (2014) no. 4, pp. 299-307

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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 $B.B = \{bb'\mid b, b' \in B\}$ cannot be greater than $O(\frac{n\log^2 n}{\log \log n})$ and present an example of a product set containing an arithmetic progression of length $\Omega(n \log n)$.For sets of complex numbers we obtain the upper bound $O(n^{3/2})$.
DOI : 10.4064/aa163-4-1
Keywords: set complex numbers size prove length longest arithmetic progression contained product set mid cannot greater frac log log log present example product set containing arithmetic progression length omega log sets complex numbers obtain upper bound

Dmitrii Zhelezov 1

1 Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg 41296 Gothenburg, Sweden 
@article{10_4064_aa163_4_1,
     author = {Dmitrii Zhelezov},
     title = {Product sets cannot contain long arithmetic progressions},
     journal = {Acta Arithmetica},
     pages = {299--307},
     publisher = {mathdoc},
     volume = {163},
     number = {4},
     year = {2014},
     doi = {10.4064/aa163-4-1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/aa163-4-1/}
}
TY  - JOUR
AU  - Dmitrii Zhelezov
TI  - Product sets cannot contain long arithmetic progressions
JO  - Acta Arithmetica
PY  - 2014
SP  - 299
EP  - 307
VL  - 163
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4064/aa163-4-1/
DO  - 10.4064/aa163-4-1
LA  - en
ID  - 10_4064_aa163_4_1
ER  - 
%0 Journal Article
%A Dmitrii Zhelezov
%T Product sets cannot contain long arithmetic progressions
%J Acta Arithmetica
%D 2014
%P 299-307
%V 163
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4064/aa163-4-1/
%R 10.4064/aa163-4-1
%G en
%F 10_4064_aa163_4_1
Dmitrii Zhelezov. Product sets cannot contain long arithmetic progressions. Acta Arithmetica, Tome 163 (2014) no. 4, pp. 299-307. doi: 10.4064/aa163-4-1

Cité par Sources :