On the diaphony of the van der Corput-Halton sequence
Let \(\sigma =(x_ n)^{\infty}_{n=0}\) be a set in the unit interval. Let \[ F_ N(\sigma)=(2\sum^{\infty}_{h=1}h^{-2}\quad | N^{- 1}\sum^{N-1}_{n=0}\exp (2\pi ihx_ n)|^ 2\quad)^{1/2}. \] \(F_ N(\sigma)\) is called the diaphony of \(\sigma\). The authors prove that \(F_ N(\sigma)\ll \sqrt{\log N}/N\) holds for the Van der Corput- Halton sequence. The estimation is best possible, since we know that for any set \(\sigma\), \(F_ N(\sigma)>(1/68)(\sqrt{\log N}/N)\) holds for infinitely many N.
scientific article; zbMATH DE number 4067036
scientific article