Problem: Given three positive integers $0 < n_1 < n_2 < n_3$ such that $$n_1 + n_2 \ne n_3, \quad n_2 \ne 2n_1, \quad n_3 \ne 2n_1, \quad n_3 \ne 2n_2,$$ is there always a real number $x$ such that $$\cos n_1 x + \cos n_2 x + \cos n_3 x < -2?$$

This is a follow-up of the question in MO: Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?

I checked that (brute force), the statement is true for all positive integers $1\le n_1 < n_2 < n_3 \le 1000$ with $n_1 + n_2 \ne n_3, \, n_2 \ne 2n_1, \, n_3 \ne 2n_1, \, n_3 \ne 2n_2$. For $n_3 > 1000$, I randomly generate some $n_1, n_2, n_3$ without finding a counterexample.

I also found a related problem in MO:
The maximum of a real trigonometric polynomial.

Perhaps the problem here is easier to deal with.

My idea is to consider the case $\cos n_1 x = -1$ or
$\cos n_2 x = -1$ or $\cos n_3 x = -1$.

For example, $\cos n_3 x = -1$ leads to $x = \frac{(2k_3 + 1)\pi}{n_3}$ where $0\le k_3 \le n_3; k_3\in \mathbb{Z}$;

We consider $f(k_3) = \cos \frac{(2k_3 + 1)n_1\pi}{n_3} + \cos\frac{(2k_3 + 1)n_2\pi}{n_3} - 1$. However, I failed to go proceed.

I also tried non-negative trigonometric polynomials
and found this article: "Extremal Positive Trigonometric Polynomials", https://www.dcce.ibilce.unesp.br/~dimitrov/papers/main.pdf

Theorem 4 gives the necessary and sufficient condition for
$f(\theta) = a_0/2 + \sum_{k=1}^n a_k \cos k \theta$ to be non-negative on $\mathbb{R}$.

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