Erdős–Turán-Type Theorems on Piecewise Smooth Curves and Arcs Original Research Article

IfLis a Jordan curve or a Jordan arc andpnis a monic polynomial of degreenwe obtain estimates for the discrepancy between the equilibrium measureμLofLand the distributionνpnof the zeros ofpnbased on one-sided bounds for the differenceU(μL−νpn, z) of their logarithmic potentials. These new estimates generalize known results to the case thatLis not smooth, i.e., corners ofLare allowed, but cusps are not. Moreover, the results are independent of the angles at the corners. The method of proof shows that both situations—upperorlower bounds ofU(μL−νpn, z)—can be treated simultaneously. As an application, the distribution of Fekete points and extremal points of best uniform approximants can be investigated generalizing results of Kleiner [14] and Blatt and Grothmann [6] to Jordan curves and arcs with corners.