Cosine Sign Correlation

Fix $\{a_1,\dots ,a_n\}\subset \mathbb{N}$, 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 $\mathbb{P}(a_1,a_2)\ge 1\mathrm{/}3$ with equality if and only if $\{a_1,a_2\}=\gcd (a_1,a_2)\cdot \{1,3\}$. We prove $\mathbb{P}(a_1,a_2,a_3)\ge 1\mathrm{/}9$ with equality if and only if $\{a_1,a_2,a_3\}=\gcd (a_1,a_2,a_3)\cdot \{1,3,9\}$. The pattern does not continue, as $\{1,3,11,33\}$ achieves a smaller value than $\{1,3,9,27\}$. We conjecture multiples of $\{1,3,11,33\}$ to be optimal for $n=4$, discuss implications for eigenfunctions of Schrödinger operators $-\mathrm{\Delta }+V$, and give an interpretation of the problem in terms of the lonely runner problem.