Resnikoff silver numbers and tilings of the half-line (Dedicated to the memory of H.L.Resnikoff)

TL;DR

This paper explores the properties of silver numbers, generalizations of the golden number, and their role in constructing inflationary tilings of the half-line, extending known results for the golden number to a broader class.

Contribution

It generalizes the connection between silver numbers and inflationary tilings, providing conditions under which such tilings exist for arbitrary silver numbers.

Findings

Tilings are obtained if differences of silver integers satisfy a non-accumulation condition.

The paper characterizes when silver integers lead to inflationary tilings.

Provides a detailed proof of a non-periodicity result related to Penrose tilings.

Abstract

Building on work by H.L.Resnikoff we consider (Resnikoff) silver numbers, which generalize the familiar golden number. By definition, a silver number is the largest positive root of a certain polynomial called silver polynomial. In turn, a corresponding companion matrix of a silver polynomial gives rise to a well known construction of inflationary tilings of the (non-negative) real half-line, via an iteration of inflation and substitution. Resnikoff noted for the golden number $\phi$ that this tiling corresponds to the set of what he called $\phi$-integers. We generalize this result for a special class of silver numbers, the distinguished silver numbers, by showing that the integers for a distinguished silver number give rise to a tiling, of which we provide a precise description. For the general problem, whether the integers for an arbitrary silver number give rise to a tiling, we cannot give a general answer, but we show that tilings are obtained if and only if the differences of silver integers satisfy a (rather weak looking) non-accumulation condition. If tilings of this type exist for certain (necessarily non-distinguished) silver numbers, they would seem to form a class of inflationary tilings that differs from those obtained by inflation and substitution. In an Appendix we recall necessary notions and -- mostly known -- results, including the inflation-substitution construction principle for (one dimensional) inflationary tilings, in an elementary manner. For the readers' convenience we also collect the pertinent facts about non-negative matrices, thus the construction is accessible with only basic prerequisites from linear algebra and analysis. Finally, in our setting we give a detailed proof of a non-periodicity result that goes back to Penrose.