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We give an overview on discrepancy theorems based on bounds of the logarithmic potential of signed measures. The results generalize well-known results of P. Erdős and P. Turán on the distribution of zeros of polynomials. Besides of new estimates for the zeros of orthogonal polynomials, we give further applications to approximation theory concerning the distribution of Fekete points, extreme points and zeros of polynomials of best uniform approximation.

Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy $E_{\rho \left(f\right)}$, ρ(f) < ∞. We investigate rational approximants $r_{n,mₙ}$ of f on E with numerator degree ≤ n and denominator degree ≤ mₙ. We show that a geometric convergence rate of order $\rho {\left(f\right)}^{-n}$ on E implies uniform maximal convergence in m₁-measure inside $E_{\rho \left(f\right)}$ if mₙ = o(n/log n) as n → ∞. If mₙ = o(n), n → ∞, then maximal convergence in capacity inside $E_{\rho \left(f\right)}$ can be proved at least for a subsequence Λ ⊂ ℕ. Moreover, an analogue of Walsh’s...