Cosine Sign Correlation

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.