Spectral gap and sign changes of Gaussian stationary processes

It is known that the Fourier transform of a measure which vanishes on [-a,a]
must have asymptotically at least a/pi zeroes per unit interval.
One way to quantify this further is using a probabilistic model:
Let f be a Gaussian stationary process on R whose spectral measure vanishes on [-a,a].
What is the probability that it has no zeroes on an interval of length L?
Our main result shows that this probability is at most e^{-c a^2 L^2}, where c>0 is an absolute constant.
This settles a question which was open for a while in the theory of Gaussian processes.
I will explain how to translate the probabilistic problem to a problem of minimizing weighted
L^2 norms of polynomials against the spectral measure, and how we solve it using tools from
harmonic and complex analysis. Time permitting, I will discuss lower bounds.
Based on a joint work with Ohad Feldheim, Benjamin Jaye, Fedor Nazarov and Shahaf Nitzan (arXiv:1801.10392).