arXiv subjects

James Rickards

Publications and source records attributed to James Rickards.

12 recordsLinked to original sources

arXiv2026

Eisenstein circle packings and the Eisenpint Schmidt arrangement

The Schmidt arrangement of an imaginary quadratic number field $K$ is the orbit of the extended real line under $\text{PSL}(2, \mathcal{O}_K)$ as M\"obius transformations on the extended complex plane. If $K\neq\mathbb{Q}(\sqrt{-3})$, then the resulting set of circles can only intersect tangentially, leading to various classes of integral circle packings, including Apollonian circle packings. When $K=\mathbb{Q}(\sqrt{-3})$, circles can intersect at angles of $\frac{\pi}{3}$ and $\frac{2\pi}{3}$, making it unclear how to extract circle packings from the arrangement. The goal of this paper is to study a modification of the $\mathbb{Q}(\sqrt{-3})-$Schmidt arrangement called the "Eisenpint Schmidt arrangement" and associated integral "Eisenstein circle packings". In analogy to the study of Apollonian circle packings, we study the number theory of such packings, including associated families of quadratic forms, show the Eisenpint Schmidt arrangement is formed of exactly all primitive Eisenstein circle packings, show strong approximation and classify congruence obstructions, prove a density-one local-global statement, and find quadratic -- but alas no cubic -- reciprocity obstructions. Unexpected aspects of the Eisenstein case include the role of congruence subgroups, the bipartite nature of the packings and reciprocity obstructions, the coefficients of quadratic obstructions, an abundance of extra symmetry, and the need to use "first-odd" quadratic forms.

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arXiv2025

An Interval Type-2 Version of Bayes Theorem Derived from Interval Probability Range Estimates Provided by Subject Matter Experts

Bayesian inference is widely used in many different fields to test hypotheses against observations. In most such applications, an assumption is made of precise input values to produce a precise output value. However, this is unrealistic for real-world applications. Often the best available information from subject matter experts (SMEs) in a given field is interval range estimates of the input probabilities involved in Bayes Theorem. This paper provides two key contributions to extend Bayes Theorem to an interval type-2 (IT2) version. First, we develop an IT2 version of Bayes Theorem that uses a novel and conservative method to avoid potential inconsistencies in the input IT2 MFs that otherwise might produce invalid output results. We then describe a novel and flexible algorithm for encoding SME-provided intervals into IT2 fuzzy membership functions (MFs), which we can use to specify the input probabilities in Bayes Theorem. Our algorithm generalizes and extends previous work on this problem that primarily addressed the encoding of intervals into word MFs for Computing with Words applications.

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arXiv2025

Primes represented by shifted quadratic forms: on primitivity and congruence classes

We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an arithmetic progression. This extends the sieve methods of Iwaniec, who showed such lower bounds without the primitivity and congruence conditions. Imposing primitivity adds some subtleties to the local criteria for representation of a shifted prime: for example, some shifted quadratic forms of discriminant $5 \pmod{8}$ do not primitively represent infinitely many primes. We also provide a careful list of the local conditions under which a genus of an integral binary quadratic form represents an integer, verified by computer, and correcting some minor errors in previous statements. The motivation for this work is as a tool for the study of prime components in Apollonian circle packings [FFH+24]

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arXiv2024

Prime and thickened prime components in Apollonian circle packings

Inspired by a question of Sarnak, we introduce the notion of a prime component in an Apollonian circle packing: a maximal tangency-connected subset having all prime curvatures. We also consider thickened prime components, which are augmented by all circles immediately tangent to the prime component. In both cases, we ask about the curvatures which appear. We consider the residue classes attained by the set of curvatures, the number of circles in such components, the number of distinct integers occurring as curvatures, and the number of prime components in a packing. As part of our investigation, we computed and analysed example components up to around curvature $10^{13}$; software is available.

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arXiv2024

Reciprocity obstructions in semigroup orbits in SL(2, Z)

We study orbits of semigroups of $\text{SL}(2,\mathbb{Z})$, and demonstrate reciprocity obstructions: we show that certain such orbits avoid squares, but not as a consequence of obstructions inherited from an algebraic set, and not as a consequence of congruence obstructions. This is in analogy to the reciprocity obstructions recently used to disprove the Apollonian local-global conjecture. We give an example of such an orbit which is known exactly, and misses all squares together with an explicit finite list of sporadic values: the corresponding semigroup is not thin, but is dense in an algebraic variety that does not have such obstructions. We also demonstrate thin semigroups with reciprocity obstructions, including semigroups associated to continued fractions formed from finite alphabets. Zaremba's conjecture states that for continued fractions with coefficients chosen from $\{1,\ldots,5\}$, every positive integer appears as a denominator. Bourgain and Kontorovich proposed a generalization of Zaremba's conjecture in the context of semigroups associated to finite alphabets. We disprove their conjecture. In particular, we demonstrate classes of finite continued fraction expansions which never represent rationals with square denominator, but not as a consequence of congruence obstructions, and for which the limit set has Hausdorff dimension exceeding $1/2$. An example of such a class is continued fractions of the form $[0; a_1, a_2, \ldots, a_n,1,1,2]$, where the $a_i$ are chosen from the set $\{4,8,12,\ldots,128\}$. The object at the heart of these results is a semigroup $\Psi\subseteq\Gamma_1(4)$ which preserves Kronecker symbols.

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arXiv2023

The local-global conjecture for Apollonian circle packings is false

In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.

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arXiv2023

Linking number of modular knots

We compute the linking number of two modular knots in the space $\text{PSL}(2, \mathbb{Z})\backslash\text{PSL}(2, \mathbb{R})$ with the trefoil filled in, which answers a question posed by Ghys in 2007. This computation is realized through the correspondence between modular links and Lorenz links, and can be thought of as an intersection number involving Conway topographs. We compare this to a second formula for the linking number of Lorenz links, which was proven by Stephen F. Kennedy in 1994.

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arXiv2022

The Apollonian staircase

A circle of curvature $n\in\mathbb{Z}^+$ is a part of finitely many primitive integral Apollonian circle packings. Each such packing has a circle of minimal curvature $-c\leq 0$, and we study the distribution of $c/n$ across all primitive integral packings containing a circle of curvature $n$. As $n\rightarrow\infty$, the distribution is shown to tend towards a picture we name the Apollonian staircase. A consequence of the staircase is that if we choose a random circle packing containing a circle $C$ of curvature $n$, then the probability that $C$ is tangent to the outermost circle tends towards $3/\pi$. These results are found by using positive semidefinite quadratic forms to make $\mathbb{P}^1(\mathbb{C})$ a parameter space for (not necessarily integral) circle packings. Finally, we examine an aspect of the integral theory known as spikes. When $n$ is prime, the distribution of $c/n$ is extremely smooth, whereas when $n$ is composite, there are certain spikes that correspond to prime divisors of $n$ that are at most $\sqrt{n}$.

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arXiv2021

Improved computation of fundamental domains for arithmetic Fuchsian groups

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.

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arXiv2021

Hecke operators acting on optimal embeddings in indefinite quaternion algebras

We explore a natural action of Hecke operators acting on formal sums of optimal embeddings of real quadratic orders into Eichler orders. By associating an optimal embedding to its root geodesic on the corresponding Shimura curve, we can consider the signed intersection number of pairs of embeddings. Using the Hecke operators and the intersection pairing, we construct a generating series that is demonstrated to be a classical modular form of weight two.

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arXiv2021

Counting intersection numbers of closed geodesics on Shimura curves

Let $\Gamma\subseteq\text{PSL}(2, \mathbb{R})$ correspond to the group of units of norm $1$ in an Eichler order $\mathrm{O}$ of an indefinite quaternion algebra over $\mathbb{Q}$. Closed geodesics on $\Gamma\backslash\mathbb{H}$ correspond to optimal embeddings of real quadratic orders into $\mathrm{O}$. The weighted intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon-Vonk on a real quadratic analogue to the difference of singular moduli. In this paper, we study the total intersection number over all embeddings of a given pair of discriminants. We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection. This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing the difference of singular moduli. The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper.

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arXiv2019

Computing intersections of closed geodesics on the modular curve

In a recent work of Duke, Imamo\={g}lu, and T\'{o}th, the linking number of certain links on the space $\text{SL}(2,\mathbb{Z})\backslash\text{SL}(2,\mathbb{R})$ is investigated. This linking number has an alternative interpretation as the intersection number of closed geodesics on the modular curve, which is the focus of this paper. By relating the intersection number to a combinatorial computation involving rivers of Conway topographs, an efficient algorithm for computing intersection numbers is produced. A formula for the total intersection of a pair of positive coprime fundamental discriminants is also derived, which can be thought of as a real quadratic analogue of a classical result of Gross and Zagier. The paper ends with numerical computations and distribution questions relating to intersection numbers.

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