arXiv subjects

Margaret Stanier

Publications and source records attributed to Margaret Stanier.

3 recordsLinked to original sources

arXiv2026

Optimal local convergence criteria for integer and Gaussian integer continued fractions

The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway--Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.

math.CO↗

arXiv2021

Regular coverings and parallel products of Farey maps

We examine the structure of Farey maps, which are a class of maps (graph embeddings on surfaces) that have received significant attention recently. We describe how they are related to each other through regular coverings and parallel products, and use these observations to find their complete spectra, recovering some known results. We then examine a similar class of maps defined by Hecke groups.

math.CO↗

arXiv2021

Necessary and sufficient conditions for convergence of integer continued fractions

Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational. An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is a graph embedded in the hyperbolic plane that induces a tessellation of the hyperbolic plane by ideal triangles. With this geometric representation of continued fractions our test for convergence can be interpreted in a particularly elegant manner, giving deeper insight into the nature of continued fraction convergence.

math.NT↗