Generalized Erd\H{o}s-Tur\'an inequalities and stability of energy minimizers

TL;DR

This paper extends Erd ext{"o}s-Turán inequalities to higher dimensions using Riesz potentials and Wasserstein-infinity distance, providing new bounds and stability results for energy minimizers in potential theory.

Contribution

It generalizes classical inequalities from 1D to higher dimensions with Riesz potentials and introduces Wasserstein-infinity distance to quantify measure proximity.

Findings

Established sharp inequalities bounding Wasserstein-infinity distance by Riesz potential norms.

Proved the inequalities are sharp up to constants for singular Riesz potentials.

Applied inequalities to demonstrate stability of energy minimizers with clustering behavior.

Abstract

The classical Erd\H{o}s-Tur\'an inequality on the distribution of roots for complex polynomials can be equivalently stated in a potential theoretic formulation, that is, if the logarithmic potential generated by a probability measure on the unit circle is close to $0$, then this probability measure is close to the uniform distribution. We generalize this classical inequality from $d=1$ to higher dimensions $d>1$ with the class of Riesz potentials which includes the logarithmic potential as a special case. In order to quantify how close a probability measure is to the uniform distribution in a general space, we use Wasserstein-infinity distance as a canonical extension of the concept of discrepancy. Then we give a compact description of this distance. Then for every dimension $d$, we prove inequalities bounding the Wasserstein-infinity distance between a probability measure $\rho$ and the uniform distribution by the $L^p$-norm of the Riesz potentials generated by $\rho$. Our inequalities are proven to be sharp up to the constants for singular Riesz potentials. Our results indicate that the phenomenon discovered by Erd\H{o}s and Tur\'an about polynomials is much more universal than it seems. Finally we apply these inequalities to prove stability theorems for energy minimizers, which provides a complementary perspective on the recent construction of energy minimizers with clustering behavior.