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 ${\mathbb {P}}(a_1,\ldots ,a_n)$ 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 ${\mathbb {P}}(a_1,a_2) \ge 1/3$ 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.