Erdős-Turán inequalities for distance functions on spheres
Let \(w_ N\) be an \(N\)-point subset of the unit circle \(| z|=1\) and let \(D(w_ N)\) denote its discrepancy. A classical result of \textit{P. Erdős} and \textit{P. Turan} [Ann. Math., II. Ser. 41, 162-173 (1940; Zbl 0023.02201)] relates the discrepancy to the maximum modules \(M(w_ N)\) of the corresponding polynomial \(p(z,w_ N)\) on the unit circle, that is, the monic polynomial, whose zeros are points of \(w_ N\): \[ M(w_ N)\geq\exp(cD^ 2(w_ N)/N) \] with some constant \(c>0\). In the present paper similar inequalities are proved for a large class of distance functions defined for finite subsets \(w_ N\) of the unit sphere \(S^{d- 1}\) in \(d\)-dimensional Euclidean space \((d\geq 2)\). These inequalities relate the so called ``spherical cap discrepancy'' to certain ``potentials'' and ``energy sums'' generated by \(w_ N\). Furthermore an improvement of the classical inequality is obtained by studying the \(L^ 1\)-norm of the function \(\log| p(z,w_ N)|\) instead of its maximum.
scientific article; zbMATH DE number 63007