Cosine Sign Correlation

Problem's Description

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of $n$ distinct positive integers. Let $X$ be a uniformly distributed random variable on $[0,2\pi]$. How small can the probability

$$p_n = \inf_{a_1, a_2, \dots, a_n} \mathbb{P}\left( \cos(a_1 x), \cos(a_2 x), \cos(a_3 x), \dots, \cos(a_n x) \qquad \mbox{all have the same sign}\right)$$ be? This question arises naturally when looking at sign correlations of eigenfunctions of Schrodinger operators. We know that $$ p_2 = \frac{1}{3}$$ which appeared as a Lemma in Goncalves, Oliveira e Silva, Steinerberger (2019). An extremal set is given by $\left\{a_1, a_2\right\} = \left\{1, 3\right\}$. A more recent result is $$ p_3 = \frac{1}{9}$$ which was shown by Dou-Goh-Liu-Legate-Pettigrew (2024), the extremal set is $\left\{a_1, a_2, a_3\right\} = \left\{1, 3, 9\right\}$ They also prove that $$ p_n \leq \frac{1}{3^{n-1}}$$ by considering the powers of 3 as an example. This, however, is not the end of the story: they also show that $$ p_4 \leq \frac{1}{33}$$ with an example given by $1,3,11,33$ and $$ p_5 \leq \frac{1}{105}$$ with an example being given by $1, 3, 11, 35, 105$. How quickly does $p_n$ decay and what can be said about the extremal sets?

1. Article Cosine Sign Correlation

   • S. Dou,
   • A. Goh,
   • K. Liu,
   • M.Legate,
   • G. Pettigrew

   Journal of Fourier Analysis and Applications, 2024arXiv

2. Article A universality law for sign correlations of eigenfunctions of differential operators

   • F. Goncalves,
   • D. Oliveira e Silva,
   • S. Steinerberger

   Journal of Spectral Theory, 2019arXiv