On regularities of the distribution of special sequences

Abstract

Ifp≥2 is an integer, then every nonnegative integerk is represented by an expression of the form $$k = \sum\nolimits_{i = 0}^s {a_i (k)p^i } $$ with integersa i (k), 0≤a i (k)≤p−1,i=0.1,...,s. The “radical-inverse function” to the basep,Φ p (k), is defined by $$\Phi _p (k) = \sum\nolimits_{i = 0}^s {a_i (k)p^{ - i - 1} } $$ . The sequence $$\omega = (\Phi _p (k))_{k = 0}^\infty $$ is uniformly distributed modulo 1 (it may be called a one-dimensional Halton sequence). In the casep=2 it is the van der Corput sequence. The set of all numbers β∈(0, 1] such that the “local” discrepancy $$\sum\nolimits_{k = 0\chi _{\left[ {0\beta } \right)} }^{n - 1} {\left( {\Phi _p (k)} \right) - } n\beta $$ is bounded inn is determined.