Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $U$ be a uniform distribution in the interval $I$. I am interested in the quantity $\mathbb{P}(f(U)>0)$.

Questions: 1) (strong form) How large should $\delta(n)$ so that we would have $\mathbb{P}(f(U)>0)\approx 1/2$?

2. (weak form) How large should $\delta$ so that we would have $\mathbb{P}(f(U)>0)$ is bounded away from zero and one independently of $n$?

Comment: If $s_n$ is the Dirichlet kernel, that is, $k_i=i$, it is easy to see that we must have $\delta_n>>n^{-1}$. I would be content if one of the latter statements was true with $\delta(n)=1/\log (n)$.

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a uniform distribution in the interval $I$. I am interested in the quantity $\mathbb{P}(s_n(U)>0)$.

Questions: 1) (strong form) How large should $\delta(n)$ be so that we would have $\mathbb{P}(s_n(U)>0)\approx 1/2$?

2. (weak form) How large should $\delta$ be so that we would have $\mathbb{P}(s_n(U)>0)$ is bounded away from zero and one independently of $n$?

Comment: If $s_n$ is the Dirichlet kernel, that is, $k_i=i$, it is easy to see that we must have $\delta_n>>n^{-1}$. I would be content if one of the latter statements was true with $\delta(n)=1/\log (n)$.