The maximal number of $3$-term arithmetic progressions in finite sets in different geometries

TL;DR

This paper investigates the maximum number of 3-term arithmetic progressions in various metric spaces, extending known results from Euclidean spaces to certain manifolds and showing limitations in spherical and tree geometries.

Contribution

It generalizes the maximal count of 3-term arithmetic progressions to Cartan--Hadamard manifolds and identifies geometries where the Euclidean result does not hold.

Findings

Maximal number extends to Cartan--Hadamard manifolds and hyperbolic spaces.

The result does not hold in spherical geometry.

The result does not hold in r-regular trees for r ≥ 3.

Abstract

Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any Euclidean space. We study this problem in general metric spaces, where a triple $(a,b,c)$ of points in a metric space is considered a $3$-term arithmetic progression if $d(a,b)=d(b,c)=\frac{1}{2}d(a,c)$. In particular, we show that the result of Green and Sisask extends to any Cartan--Hadamard manifold (in particular, to the hyperbolic spaces), but does not hold in spherical geometry or in the $r$-regular tree, for any $r\geq 3$.