Cosine Sign Correlation

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 $\mathbb{P}(a_1,\dots ,a_n)$ 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 $\mathbb{P}(a_1,a_2)\ge 1/3$ with equality if and only if $\left\{a_1,a_2\right\}=\gcd (a_1,a_2)\cdot \left\{1,3\right\}$. We prove $\mathbb{P}(a_1,a_2,a_3)\ge 1/9$ 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 Schrödinger operators $-\Delta +V$, and give an interpretation of the problem in terms of the lonely runner problem.

9 pages, 1 figure