Sums Associated with Orbits in the Binary Dynamical System

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 the unit interval \([0,1)\) 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, in part, to the behaviour that was observed by Hardy and Littlewood in a different context. A number in \([0,1)\) is called a binary rational if it is of the form \(k/2^n\) for some \(0\leqslant k<n\). Given a number in \([0,1)\) that is not a binary rational, its sequence of binary digits may be arranged as an infinite sequence of consecutive, non-empty, finite blocks, each block consisting of all zeros or all ones. Upper and lower estimates are derived for the sums, and for the averages of 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. In each case, a necessary and sufficient condition on the blocks is found for the 'equivalence' of the upper and lower estimates. In the case of the averages, this is a semi-symmetric 'regularity' condition on the blocks. The numbers in \([0,1)\) that satisfy the regularity condition have measure \(1\), and include all base two simply normal numbers and all algebraic numbers that are not binary rationals. If a number is not a binary rational and does not satisfy the regularity condition then it is a Liouville number, and there is an uncountable set of such numbers.