A Bombieri-type inequality and equidistribution of points

TL;DR

This paper extends Bombieri-type inequalities to geometric settings like the sphere and torus, introduces pseudopolynomials, and explores their implications for point distribution and packing problems on these surfaces.

Contribution

It generalizes Bombieri inequalities using geometric interpretations, introduces pseudopolynomials on the torus, and connects these to packing and distribution problems on Riemann surfaces.

Findings

Derived sharp Bombieri-type inequalities for sphere and torus.

Introduced pseudopolynomials based on Weierstrass sigma function.

Established bounds related to packing numbers and point configurations.

Abstract

In recent work, Etayo introduces a new Bombieri-type inequality for monic polynomials. Here we reinterpret this new inequality as a more general integral inequality involving the Green function for the sphere. This rather geometric interpretation allows for generalizations of the basic inequality, involving fractional zeros while also opening up the possibility to extend the setting to general compact Riemann surfaces. We derive a sharp form of these generalized Bombieri-type inequalities for the case of the sphere and the torus. These inequalities involve a quantity we call the packing number, which in turn is inspired by the geometric zero packing problems considered by Hedenmalm in the context of the asymptotic variance of the Bergman projection of a bounded function. As for the torus, we introduce analogs of polynomials (pseudopolynomials) based on the classical Weierstrass $\sigma$ function, and we explain how such pseudopolynomials fit in with the extended geometric Bombieri-type inequality. The sharpness of the packing number bound on the torus involves the construction of a lattice configuration on the torus for any given integer number of points. The corresponding bound for the sphere instead relies on the existence of well-conditioned polynomials in the sense of Shub and Smale.