Averaging sets on the unit circle (Q1335048)

Let \(\sigma\) be an arbitrary normalized measure on the unit circle \(T\). Let \(t_ \sigma (n)\) be the maximal integer \(t\) for which the quadrature formula of Chebyshev type given by the formula \[ \int p(x,y)d\sigma= {\textstyle {1\over n}} \sum_{k=1}^ n p(x_ k, y_ k) \] holds for some subset \(\{(x_ 1,y_ 1), (x_ 2,y_ 2), \dots, (x_ n,y_ n)\}\) of \(T\) and also for all polynomials \(p(x,y)\) of degree \(p\leq t\). Let \(\omega\) be the Lebesgue measure then \(t_ \omega(n)= n-1\). Further \(t_ \sigma(n)\leq n-1\) for every \(\sigma\). By using the Kolmogorov- Szegő condition on \(\sigma\) it is proved that \(\sigma= \omega\) if \(t_ \sigma(n)= n-1\) for a subsequence of \(n=1,2,\dots\).