**Problem**: Given three positive integers $0 < n\_1 < n\_2 < n\_3$. Is there always a real number $x$ such that
$$\cos n\_1 x + \cos n\_2 x + \cos n\_3 x < -2?$$



Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
The answer is negative. If $n\_3=n\_1+n\_2$, then
$$\cos n\_1 x + \cos n\_2 x + \cos n\_3 x\geq -2.$$
Indeed, the left-hand side equals
$$(1+\cos n\_1 x)(1+\cos n\_2 x)-1-\sin n\_1 x\sin n\_2 x,$$
where $1+\cos n\_j x\geq 0$, and $\sin n\_1 x\sin n\_2 x\leq 1$.
Furthermore, if $n\_2=2 n\_1$ and $n\_3=3 n\_1$, then
$$\min\_{x\in\mathbb{R}}\,(\cos n\_1 x + \cos n\_2 x + \cos n\_3 x)=-\frac{17+7\sqrt{7}}{27}\approx-1.31557.$$
More generally, the minimum of the [Dirichlet kernel](https://en.wikipedia.org/wiki/Dirichlet_kernel) is known to great precision, see e.g. [here](http://www.idmercer.com/dirichletkernel.pdf).


