Cosine Sign Correlation

TL;DR

This paper investigates the minimal probability that cosine functions of integer multiples of a random angle are all positive or all negative, extending previous results and exploring implications in spectral theory and the lonely runner problem.

Contribution

It proves new lower bounds for the probability involving three cosine functions and explores the pattern's failure for four functions, proposing conjectures and applications.

Findings

Proved /9 as a lower bound for three cosine functions.

Identified specific sets /9 that achieve equality for three functions.

Discovered the pattern breaks for four functions, with a conjecture on optimal sets.

Abstract

Fix $\left\{a_1, \dots, a_n \right\} \subset \mathbb{N}$, and let $x$ be a uniformly distributed random variable on $[0,2\pi]$. The probability $\mathbb{P}(a_1,\ldots,a_n)$ that $\cos(a_1 x), \dots, \cos(a_n x)$ are either all positive or all negative is non-zero since $\cos(a_i x) \sim 1$ for $x$ in a neighborhood of $0$. We are interested in how small this probability can be. Motivated by a problem in spectral theory, Goncalves, Oliveira e Silva, and Steinerberger proved that $\mathbb{P}(a_1,a_2) \geq 1/3$ with equality if and only if $\left\{a_1, a_2 \right\} = \gcd(a_1, a_2)\cdot \left\{1, 3\right\}$. We prove $\mathbb{P}(a_1,a_2,a_3)\geq 1/9$ with equality if and only if $\left\{a_1, a_2, a_3 \right\} = \gcd(a_1, a_2, a_3)\cdot \left\{1, 3, 9\right\}$. The pattern does not continue, as $\left\{1,3,11,33\right\}$ achieves a smaller value than $\left\{1,3,9,27\right\}$. We conjecture multiples of $\left\{1,3,11,33\right\}$ to be optimal for $n=4$, discuss implications for eigenfunctions of Schr\"odinger operators $-\Delta + V$, and give an interpretation of the problem in terms of the lonely runner problem.