Properties of Orbits and Normal Numbers in the Binary Dynamical System

Rodney Nillsen

Published 2022-04-20Version 1

In 1930 G. H. Hardy and J. E. Littlewood derived a result concerning the rate of divergence of certain series of cosecants. In more recent terminology, their result can be interpreted as a result about the behaviour of orbits in dynamical systems arising from rotations on the unit circle. In general terms, this behaviour is related to the question of `how often' a point under successive rotations gets `close' to $1$. Now, the expansion of numbers in $[0,1)$ to the base $2$ can be associated with a different system -- the binary dynamical system. This article considers orbit behaviour in the binary system that corresponds to the behaviour that was, in effect, observed by Hardy and Littlewood in systems involving rotations. Now, except for a countable set, the sequence of binary digits of a number in $[0,1)$ may be arranged as an infinite sequence of consecutive, finite blocks, each block consisting of all zeros or all ones. The relationships between the lengths of these blocks determine Hardy-Littlewood types of behaviour in the binary system. This behaviour is considered and results relating to normal and simply normal numbers are obtained. There also are suggestions for further investigation.