1Department of Mathematics The University of Haifa at Oranim Tivon 36006, Israel 2Department of Mathematics Technion – Israel Institute of Technology Haifa 32000, Israel
Acta Arithmetica, Tome 163 (2014) no. 2, pp. 127-140
We show that if $A$ and $B$ are finite sets of real numbers, then the number of triples $(a,b,c)\in A\times B\times (A\cup B)$ with $a+b=2c$ is at most $(0.15+o(1))(|A|+|B|)^2$ as $|A|+|B|\to \infty $. As a corollary, if $A$ is antisymmetric (that is, $A\cap (-A)=\emptyset $), then there are at most $(0.3+o(1))|A|^2$ triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$. In the general case where $A$ is not necessarily antisymmetric, we show that the number of triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$ is at most $(0.5+o(1))|A|^2$. These estimates are sharp.
Keywords:
finite sets real numbers number triples times times cup infty corollary antisymmetric cap a emptyset there triples a b general where necessarily antisymmetric number triples a b these estimates sharp
Affiliations des auteurs :
Vsevolod F. Lev
1
;
Rom Pinchasi
2
1
Department of Mathematics The University of Haifa at Oranim Tivon 36006, Israel
2
Department of Mathematics Technion – Israel Institute of Technology Haifa 32000, Israel
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author = {Vsevolod F. Lev and Rom Pinchasi},
title = {Solving $a\pm b=2c$ in elements of finite sets},
journal = {Acta Arithmetica},
pages = {127--140},
year = {2014},
volume = {163},
number = {2},
doi = {10.4064/aa163-2-3},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa163-2-3/}
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Vsevolod F. Lev; Rom Pinchasi. Solving $a\pm b=2c$ in elements of finite sets. Acta Arithmetica, Tome 163 (2014) no. 2, pp. 127-140. doi: 10.4064/aa163-2-3
