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

TL;DR

This paper establishes sharp upper bounds on the number of solutions to the equation a ± b = 2c within finite sets, revealing asymptotic limits and special cases like antisymmetric sets.

Contribution

It provides the first asymptotic bounds for the number of solutions to a ± b = 2c in finite sets, including sharp constants and special cases.

Findings

Maximum of (0.15+o(1))( |A|+|B|)^2 solutions for A,B finite sets

At most (0.3+o(1))|A|^2 solutions when A is antisymmetric

At most (0.5+o(1))|A|^2 solutions in the general case

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.