On a weighted sum of distances from a well distributed set of points

Abstract

Suppose that ξ is a complex number, t > 0 and w1, . . . ,wd 0. Let Δ be the modulus of the product of d(d − 1)/2 distances between complex numbers z1, . . . ,zd labelled so that |z1 −ξ | . . . |zd −ξ |. We prove that the sum 1 d ∑i=1 wi|zi −ξ |t is at least √ e 2 e−1/dΔ2t/d(d−1)d−t/(d−1) d−1 ∏ i=1 w2(d−i)/d(d−1) i and show that this inequality is sharp for certain choice of weights wi. This inequality is then applied to sets of conjugate algebraic integers. Mathematics subject classification (2010): 11R04, 52A40.