  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.
