# Some Remarks on Systems of Equiangular Lines

## TL;DR

This paper investigates the maximum size of equiangular line systems in real space with a specific cosine condition, motivated by prior work and focusing on cases where the reciprocal of the cosine is not an odd positive integer.

## Contribution

It provides new insights into the bounds of equiangular line systems under certain cosine constraints, extending previous results in the field.

## Key findings

- Established bounds for N_α(d) when 1/α is not an odd positive integer
- Connected the problem to prior conjectures and results in equiangular lines
- Clarified conditions under which maximum configurations are achieved

## Abstract

In this note, we study the maximum number $N_\alpha(d)$ of a system of equiangular lines in $\mathbb{R}^d$ with cosine $\alpha$, where $\frac{1}{\alpha}$ is not an odd positive integer. This note is motivated by a remark in a $2018$ paper by Balla, Dr\"{a}xler, Keevash and Sudakov.