Equidistribution of Points via Energy

We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin’s constant, and problems on the means of zeros of polynomials with integer coefficients. 1. Asymptotic equidistribution of discrete sets Let E be a compact set in the complex plane C. Given a set of points Zn = {zk,n}k=1 ⊂ C, n ≥ 2, the associated Vandermonde determinant is V (Zn) := ∏ 1≤j<k≤n (zj,n − zk,n). Let the n-th diameter of E be defined by δn(E) := max Zn⊂E |V (Zn)| 2 n(n−1) . A set of points Fn is called the n-th Fekete points of E if it achieves the above maximum. The classical result of Fekete [15] states that δn(E), n ≥ 2, form a decreasing sequence that converges to a limit called the transfinite diameter δ(E). Szegő [40] found that δ(E) is equal to the logarithmic capacity cap(E) from potential theory, which is defined as follows. For a Borel measure μ with compact support, define its energy by [43, p. 54] I[μ] := ∫∫ log 1 |z − t| dμ(t)dμ(z). Consider the problem of finding the minimum energy VE := inf μ∈M(E) I[μ], where M(E) is the space of all positive unit Borel measures supported on E. The capacity of E is given by cap(E) := e−VE . If Robin’s constant VE is finite (i.e. cap(E) 6= 0), then the infimum is attained by the equilibrium measure μE ∈M(E) [43, p. 55], which is a unique probability measure expressing the 2000 Mathematics Subject Classification. Primary 31C20; Secondary 30C15, 31C15, 11C08.