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. One result was that for certain real numbers $\theta$, there is $K>0$ such that \[\sum_{k=1}^n\frac{1}{sin^2(k\theta\pi)}\le Kn^2, \ {\rm for\ all}\ n\in {\mathbb N}. \] In more recent terminology, this can be interpreted as a result about 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 the binary system $([0,1),f)$, where $f:[0,1)\longrightarrow [0,1)$ is given by \[ f(x)= \begin{cases} 2x, \ {\rm for}\ 0\le x<1/2,\\ 2x-1, {\rm for}\ 1/2\le x<1.\\ \end{cases} \] A number in $[0,1)$ is called a binary rational if it is not of the form $k/2^n$ for some $0\le k<2^n$, and $f^k$ denotes the $k^{th}$ iterate of $f$. Given a number in $[0,1)$ that is not a binary rational, and given $p>0$, we find upper and lower estimates for each of \[\sum_{k=1}^n\frac{1}{f^{k-1}(x)^p} \ {\rm and}\ \frac{1}{n}\left(\sum_{k=1}^n\frac{1}{f^{k-1}(x)^p}\right),\] where $n=1,2,3,\ldots.$ These estimates are in terms of the lengths of the blocks of zeros and ones in the binary expansion of $x$. 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 right-hand exression, 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. This work has appeared in Real Analysis Exchange, 50 (1), pp. 15-52, 2025.

Discrete orbits in $\beta{\mathbb N}-{\mathbb N}$. Let ${\mathbb N}$ denote the set of natural numbers and let $\beta{\mathbb N}-{\mathbb N}$ be the Stone-Cech compactification of $\mathbb N$. The shift mapping $n\longmapsto n+1$ mapping $\mathbb N$ into $\mathbb N$ extends to a continuous mapping from $\beta{\mathbb N}$ into $\beta {\mathbb N}$ and its retriction to $\beta{\mathbb N}-{\mathbb N}$ is a homeomorphism of $\beta{\mathbb N}-{\mathbb N}$. Denote this restriction by $\sigma$. Thus $\sigma:\beta{\mathbb N}-{\mathbb N} \longrightarrow \beta{\mathbb N}-{\mathbb N}$ is a homeomorphism of $\beta{\mathbb N}-{\mathbb N}$. The paper Discrete orbits in $\beta{\mathbb N}-{\mathbb N}$, is concerned with when the obit $\{\omega, \sigma(\omega),\sigma^2(\omega),\ldots \}$ of a point $\omega\in \beta{\mathbb N}- {\mathbb N}$ is a discrete subset of $\beta{\mathbb N}- {\mathbb N}$.

Banach Limits. Banach limits are continuous invariant linear forms on the Banach space of bounded sequences. They form a compact convex set in the weak*-topology of the dual of this space. It was proved in the paper Nets of extreme Banach limits that the set of extreme points of this set is non-compact. Some related research may be found in selected publications.

Rodney Nillsen, page updated 21st January, 2026