Gleichverteilung und ein Satz von Müntz. (Uniform distribution and the theorem of Müntz) (Q1073085)

If \(p=(p_ k)\) is a strictly increasing sequence of positive integers then the author considers the discrepancy \[ D_ n^{(p)} (\omega)=\sup_{k}| (1/N) \sum^{N}_{n=1}f_ k(x_ n)- \int^{1}_{0}f_ k(x) dx\quad | \quad, \] \(f_ k(x)=x^{p_ k}\), \(\omega =(x_ n)\), \(0\leq x_ n\leq 1\). If \(\sum 1/p_ k=\infty\) then it follows from a classical result of Müntz that \(\omega\) is uniformly distributed mod 1 if and only if \(\lim_{N\to \infty}D_ N^{(p)}(\omega)=0.\) The author gives a proof for this result admitting also quantitative estimates (e.g. if \((p_ k)\) is the sequence of primes then \(D_ N\leq C \log_ 5 D_ N^{(p)}/\log_ 4 D_ N^{(p)}\), \(D_ N\) denotes the usual discrepancy).