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?

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