Dyadic diaphony of digital sequences
The author considers the dyadic diaphony \(F_{2,N}(\omega)\) [see \textit{P. Hellekalek, H. Leeb}, Acta Arith. 80, 2, 187--196 (1997; Zbl 0868.11034)] of digital \((0,s)\)-sequences \(\omega\) with \(s\in\{1,2\}\). For \((0,1)\)-sequences \(\omega\) he proves \[ (N\,F_{2,N}(\omega))^2=3\sum_{u=1}^\infty \left\| \frac{N}{2^u}\right\| ^2 \] and derives some interesting consequences. In particular he shows that the dyadic diaphony of \((0,1)\) sequences satisfies a central limit theorem. Similar for \((0,2)\)-sequences \(\omega\) the author obtains \[ (N\,F_{2,N}(\omega))^2=\frac 94\sum_{u=1}^\infty u\left\| \frac{N}{2^u}\right\| ^2. \] in this case the author cannot prove a central limit theorem, but a similar weaker theorem.
scientific article; zbMATH DE number 5302787
scientific article