Sums Associated with Orbits in the Binary Dynamical System

Abstract

In 1930, G. H. Hardy and J. E. Littlewood derived results concerning rates of divergence of certain series involving cosecants. In more recent terminology, one of their results can be interpreted in terms of the behaviour of orbits in a dynamical system that is a rotation on the unit circle. Now, the expansion of numbers in $[0,1)$ to the base $2$ can be associated with a different dynamical system -- the binary 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. Given a typical number in $[0,1)$, the sequence of its binary digits may be arranged as an infinite sequence of consecutive, non-empty, 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 associated with the number. Amongst other results, upper and lower estimates are derived for the sums of powers of the reciprocals of points in the orbit of the number. These estimates are in terms of the lengths of the associated blocks. A necessary and sufficient condition is found for the essential `equivalence' of the upper and lower estimates. Almost all numbers in $[0,1)$ satisfy this condition.