Solving $a\pm b=2c$ in the elements of finite sets

Vsevolod F. Lev; Rom Pinchasi

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arXiv · 1211.6567

Solving $a\pm b=2c$ in the elements of finite sets

Vsevolod F. Lev·Rom Pinchasi

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Abstract

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, $A\cap(-A)=\est$), 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.

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Vsevolod F. Lev, Rom Pinchasi. 2012-11-28. Solving $a\pm b=2c$ in the elements of finite sets. https://arxiv.org/abs/1211.6567

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