Sampling properties of the zeroes of the Gaussian entire function

Abstract

We study sampling properties of the zero set of the Gaussian entire function on Fock spaces. Firstly, we relax Seip and Wallst\'en's density and separation conditions for sampling sets on Fock spaces to obtain weighted inequalities for sets that are not necessarily sampling. On the probabilistic front, we estimate the number of zeroes of the Gaussian entire functions that are close to each other. We use these to prove random sampling inequalities for polynomials of degree at most $d$ using ${d}+o(d)$ points, and show that, with high probability, the sampling constants grow slower than $d^\varepsilon$ for any $\varepsilon>0$. In particular, we recover a result from Lyons and Zhai in the case of the Gaussian entire function, where it is shown that the zeroes are (almost surely) a uniqueness set for the Fock space.