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

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.