Acute sets of exponentially optimal size

TL;DR

This paper introduces a simple explicit construction of large acute sets in high-dimensional Euclidean spaces, achieving sizes close to the theoretical maximum and improving upon previous constructions.

Contribution

The authors provide a new explicit construction of acute sets of size $2^{d-1}+1$ in $ ext{R}^d$, significantly larger than prior known sets.

Findings

Constructed acute sets of size $2^{d-1}+1$ in any dimension $d$

Improved the size of known acute sets exponentially

Approached the theoretical maximum size for acute sets

Abstract

We present a simple construction of an acute set of size $2^{d-1}+1$ in $\mathbb{R}^d$ for any dimension $d$. That is, we explicitly give $2^{d-1}+1$ points in the $d$-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than $2^d$. Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order $\varphi^d$ where $\varphi = (1+\sqrt{5})/2 \approx 1.618$ is the golden ratio.