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Thomas Garrity

Publications and source records attributed to Thomas Garrity.

At least 19 recordsLinked to original sources
arXiv2026

The Natural Extension for the Triangle Map (a Multi-dimensional Continued Fraction) with An Internal Symmetry from Young Conjugation

The natural extension of the triangle map (a type of multi-dimensional continued fraction algorithm) is completely described in all possible dimensions. The motivation and inspiration for this natural extension stems from the triangle map's recent link to the classical study of integer partitions. Inspired by Young conjugation for integer partitions, we show that the natural extension has an internal symmetry and allows the natural extension to be subdivided into four natural subdomains. This appears to be new even for the classical case of the natural extension for continued fractions, namely for both the classical Gauss map and the classical Farey map.

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arXiv2024

On the Factor Complexity Associated with a Family of Multidimensional Continued Fraction Algorithms

We study the complexity of $S$-adic sequences corresponding to a family of 216 multidimensional continued fractions maps, called Triangle Partition maps (TRIP maps), with an emphasis on those with low upper bounds on complexity. Our main result is to prove that the complexity of $S$-adic sequences corresponding to the triangle map (called the $(e,e,e)$-TRIP map in this paper) has upper bound at most $3n$. Our second main result is to prove an upper bound of $2n+1$ on complexity for another TRIP map. We discuss a dynamical phenomenon, which we call ``hidden $\R^2$ behavior,'' that occurs in this map and its relationship to complexity. Combining this with previously known results and a list of counter-examples, we provide a complete list of the TRIP maps which have upper bounds on complexity of at most $3n$, except for one remaining case for which we conjecture such an upper bound to hold.

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arXiv2024

Ergodicity and Algebraticity of the Fast and Slow Triangle Maps

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension $n$ are ergodic, resolving a conjecture of Messaoudi, Noguiera and Schweiger. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

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arXiv2023

On integer partitions and continued fraction type algorithms

We show that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number $n$ into two smaller numbers, with multiplicity. We provide a complete description of how such integer partitions occur and of the conjugation for the corresponding Young shapes via the dynamics of the classical Farey tree. We use the dynamics of the Farey map to get a new formula for $p(2,n)$, the number of ways for partitioning $n$ into two smaller positive integers, with multiplicity. We then do the analogue using the additive-slow-Farey version of the Triangle map (a type of multi-dimensional continued fraction algorithm), giving us a method for producing integer partitions of a positive number $n$ into three smaller numbers, with multiplicity. However different aspects of this generalisations remain unclear.

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arXiv2020

Functional analysis behind a Family of Multidimensional Continued Fractions: Part II

This paper is a direct continuation of "Functional analysis behind a Family of Multidimensional Continued Fractions: Part I," in which we started the exploration of the functional analysis behind the transfer operators for triangle partition maps, a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This allows us now to find eigenfunctions of eigenvalue 1 for transfer operators associated with select triangle partition maps on specified Banach spaces. We proceed to prove that the transfer operators, viewed as acting on one-dimensional families of Hilbert spaces, associated with select triangle partition maps are nuclear of trace class zero. We finish by deriving Gauss-Kuzmin distributions associated with select triangle partition maps.

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arXiv2020

Functional Analysis behind a Family of Multidimensional Continued Fractions: Part I

Triangle partition maps form a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This paper begins the exploration of the functional analysis behind the transfer operator of each of these maps. We show that triangle partition maps give rise to two classes of transfer operators and present theorems regarding the origin of these classes; we also present related theorems on the form of transfer operators arising from compositions of triangle partition maps. In the next paper, Part II, we will find eigenfunctions of eigenvalue 1 for transfer operators associated with select triangle partition maps on specified Banach spaces and then proceed to prove that the transfer operators, viewed as acting on one-dimensional families of Hilbert spaces, associated with select triangle partition maps are nuclear of trace class zero. We will finish in part II by deriving Gauss-Kuzmin distributions associated with select triangle partition maps.

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arXiv2017

Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences

The Stern diatomic sequence is closely linked to continued fractions via the Gauss map on the unit interval, which in turn can be understood via systematic subdivisions of the unit interval. Higher dimensional analogues of continued fractions, called multidimensional continued fractions, can be produced through various subdivisions of a triangle. We define triangle partition-Stern sequences (TRIP-Stern sequences for short), higher-dimensional generalizations of the Stern diatomic sequence, from the method of subdividing a triangle via various triangle partition algorithms. We then explore several combinatorial results about TRIP-Stern sequences, which may be used to give rise to certain well-known sequences. We finish by generalizing TRIP-Stern sequences and presenting analogous results for these generalizations.

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arXiv2017

Generalizing the Minkowski Question Mark Function to a Family of Multidimensional Continued Fractions

The Minkowski question mark function, maping the unit interval to itself, is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus the question mark function is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.

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arXiv2015

On Gauss-Kuzmin Statistics and the Transfer Operator for a Multidimensional Continued Fraction Algorithm: the Triangle Map

The Gauss-Kuzmin statistics for the triangle map (a type of multidimensional continued fraction algorithm) are derived by examining the leading eigenfunction of the triangle map's transfer operator. The technical difficulty is finding the appropriate Banach space of functions. We also show that, by thinking of the triangle map's transfer operator as acting on a one-dimensional family of Hilbert spaces, the transfer can be thought of as a family of nuclear operators of trace class zero.

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arXiv2014

Cubic Irrationals and Periodicity via a Family of Multi-dimensional Continued Fraction Algorithms

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a, a-a^2) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u,u') with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u,u') has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. Thus these results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.

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arXiv2013

A multidimensional continued fraction generalization of Stern's diatomic sequence

Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular multidimensional continued fraction algorithm (the Farey algorithm), we will generalize the diatomic sequence to a collection of numbers that quite naturally should be called the triatomic sequence (or a two-dimensional Pascal with memory sequence). As continued fractions and the diatomic sequence can be thought of as coming from systematic subdivisions of the unit interval, this new triatomic sequence will arise by a systematic subdivision of a triangle. We will discuss some of the algebraic properties for the tri-atomic sequence.

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arXiv2012

A Generalized Family of Multidimensional Continued Fractions: TRIP Maps

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

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arXiv2012

A thermodynamic classification of pairs of real numbers via the Triangle Multi-dimensional continued fraction

A new classification scheme for pairs of real numbers is given, generalizing earlier work of the author that used continued fraction, which in turn was motivated by ideas from statistical mechanics in general and work of Knauf and Fiala and Kleban in particular. Critical for this classification are the number theoretic and geometric properties of the triangle map, a type of multi-dimensional continued fraction.

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arXiv2009

A Thermodynamic Classification of Real Numbers

A new classification scheme for real numbers is given, motivated by ideas from statistical mechanics in general and work of Knauf and of Fiala and Kleban in particular. Critical for this classification of a real number will be the Diophantine properties of its continued fraction expansion.

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