Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any $\delta\rightarrow 0$ we have

\begin{equation} \int_{-\pi}^{\pi}\mathbb{P}(|P_{n}(t)|<\delta)\,dt \approx \frac{\delta}{\sqrt{n}}? \end{equation}

For Gaussian random variables this is trivial, but for discrete ones it becomes hard when $\delta \ll n^{-1/2}$. Can the averaging effect smooth things out so that it is still true in the discrete case?