Discrépances de suites homothétiques. (Discrepancy of homothetic sequences) (Q580422)

We study the normal order of the discrepancy of the family of all sequences t\({\mathcal U}\), for a given sequence \({\mathcal U}\) of real numbers. \({\mathcal U}\) is considered as a point of the probability space \({\mathfrak Y}\) of all sequences \({\mathcal U}=(u_ n)_{n\geq 1}\) such that \(\ell_ n\leq u_ n\leq m_ n\) for all n, \((\ell_ n)_{n\geq 1}\) and \((m_ n)_{n\geq 1}\) being two fixed sequences of real numbers. Probability on \({\mathfrak Y}\) is the infinite product of uniform probabilities on \([\ell_ n,m_ n].\) Assume convergence of the series \(\sum 1/\ell_ n\). Then, with some technical condition, we have for almost all sequences \({\mathcal U}\) in \({\mathfrak Y}:\) \[ D^*_ N(t{\mathcal U})=O(\sqrt{Log m_ N} \cdot N^{- 1/2})\text{ for all } t\in {\mathbb{R}}^*\quad, \] where \(D^*_ N({\mathcal V})\) is the discrepancy of the sequence \({\mathcal V}.\) Such a result needs some estimate of the probability \(P=P(d_ N\leq \lambda N^{-1/2})\), where \(d_ N\) is the supremum of the difference \(| F_ N(x)-F(x)|\), with: F is some continuous distribution function; \(F_ n\) is an associated empirical distribution function. Here, P is near 1 for the interesting values of \(\lambda\) (by Kolmogorov- Smirnov theorem). An upper estimate of \(1-P\) is obtained by elementary ways.