Discrepancy Theorems via Energy Integrals

Abstract

If E is a compact set in ℂ with cap E >0, then the equilibrium measure μ e minimizes the energy expression

$$I[\sigma ] = \int {{U^\sigma }(z)d\sigma (z)} $$

with respect to all positive Borel measures σ supported on E with σ(E) = 1. Since I[μ e]= − log cap E, the energy I[σ] is positive for all positive Borel measures σ on E if 0 < cap E < 1. Consequently, for such E,

$$\sigma {(E)^2} \leqslant \frac{{I[\sigma ]}}{{|\log capE|}}$$

Thus, the total mass σ(E) can be estimated by its energy.