Suites dont la discrépance est comparable à un logarithme. (Sequences whose discrepancy is like a logarithm)
In [J. Number Theory, 31, 208-241 (1989; Zbl 0673.10039)] we introduced the notion of selfsimilar sequence U in the unit interval, and gave a sufficient condition for having a low discrepancy, i.e. \(N D^*_ N(U)=O(Log N).\) As an application, we give here a natural construction of sequences U such that: \[ 0<\liminf_{N\to \infty}\frac{N D^*_ N(U)}{Log N}\leq \limsup_{N\to \infty}\frac{N D^*_ N(U)}{Log N}<+\infty. \] This method cannot be used to get sequences such that N \(D^*_ N(U)/Log N\) has a finite limit (open problem).
scientific article; zbMATH DE number 4179464