arXiv subjects

Jeffrey C. Lagarias

Publications and source records attributed to Jeffrey C. Lagarias.

At least 19 recordsLinked to original sources

arXiv2026

SIC-POVMs and orders of real quadratic fields

This paper concerns SIC-POVMs and their relationship to class field theory. SIC-POVMs are generalized quantum measurements (POVMs) described by $d^2$ equiangular complex lines through the origin in $\mathbb{C}^d$. Weyl--Heisenberg SICs are those SIC-POVMs described by the orbit a single vector under a finite Weyl--Heisenberg group ${\rm WH}(d)$. We relate known data on the structure and classification of Weyl--Heisenberg SICs in low dimensions to arithmetic data attached to certain orders of real quadratic fields. For $4 \le d \le 90$, we show the number of known geometric equivalence classes of Weyl--Heisenberg SICs in dimension $d$ equals the cardinality of the ideal class monoid of the real quadratic order $\mathcal{O}_{Δ_d}$ of discriminant $Δ_d=(d+1)(d-3)$; we conjecture the equality extends to all $d \ge 4$. We prove that this conjecture implies the existence of more than one geometric equivalence class of Weyl--Heisenberg SICs for $d > 22$. We conjecture Galois multiplets of SICs are in one-to-one correspondence with the over-orders $\mathcal{O}'$ of $\mathcal{O}_{Δ_d}$ in such a way that the number of classes in the multiplet equals the ring class number of $\mathcal{O}'$. We test that conjecture against known data on exact SICs in low dimensions. We refine the class field hypothesis of Appleby, Flammia, McConnell, and Yard (arXiv:1604.06098) to predict the exact class field over $\mathbb{Q}(\sqrt{Δ_d})$ generated by the ratios of vector entries for the equiangular lines defining a Weyl--Heisenberg SIC. The refined conjectures use a recently developed class field theory for orders of number fields (arXiv:2212.09177). The refined class fields assigned to over-orders $\mathcal{O}'$ have a natural partial order under inclusion; the inclusions of these fields fail to be strict in some cases. We characterize such cases and give a table of them for $d < 500$.

math.NT↗

arXiv2026

Unit-generated orders of real quadratic fields I. Class number bounds

Unit-generated orders of a quadratic field are orders of the form $\mathcal{O} = \mathbb{Z}[\varepsilon]$, where $\varepsilon$ is a unit in the quadratic field. If the order $\mathcal{O}$ is a maximal order of a real quadratic field, then the quadratic number field is necessarily of a restricted form, being of narrow Richaud--Degert type. However, every real quadratic field contains infinitely many distinct unit-generated orders. They are parametrized as $\mathcal{O} = \mathcal{O}_{n}^{\pm}$ having quadratic discriminants $Δ(\mathcal{O}) = Δ_{n}^{+} = n^2 - 4$ (for $n \geq 3$) and $Δ(\mathcal{O}) = Δ_{n}^{-} = n^2 + 4$ (for $n \geq 1$). We show the (wide or narrow) class numbers of unit-generated orders satisfy $\log \left|{\rm Cl}(\mathcal{O})\right| \sim \log \frac{1}{2}\left|Δ(\mathcal{O})\right|$ as $\left|Δ(\mathcal{O})\right| \to \infty$, using a result of L.-K. Hua. We deduce that there are finitely many unit-generated quadratic orders of class number one and finitely many unit-generated quadratic orders whose class group is $2$-torsion. We classify all unit-generated real quadratic orders having class number one. We provide numerical lists of quadratic unit-generated orders whose class groups are $2$-torsion for $Δ\leq 10^{10}$, for both wide and narrow class groups. These lists are conjecturally complete for all $Δ$.

math.NT↗

arXiv2025

$B$-orderings for all ideals $B$ of Dedekind domains and generalized factorials

This paper extends Bhargava's theory of $\mathfrak{p}$-orderings of subsets $S$ of a Dedekind ring $R$ valid for prime ideals $\mathfrak{p}$ in $R$. Bhargava's theory defines for integers $k\ge1$ invariants of $S$, the generalized factorials $[k]!_S$, which are ideals of $R$. This paper defines $\mathfrak{b}$-orderings of subsets $S$ of a Dedekind domain $D$ for all nontrivial proper ideals $\mathfrak{b}$ of $D$. It defines generalized integers $[k]_{S,T}$, as ideals of $D$, which depend on $S$ and on a subset $T$ of the proper ideals $\mathscr{I}_D$ of $D$. It defines generalized factorials $[k]!_{S,T}$ and generalized binomial coefficients, as ideals of $D$. The extension to all ideals applies to Bhargava's enhanced notions of $r$-removed $\mathfrak{p}$-orderings, and $\mathfrak{p}$-orderings of order $h$.

math.AC↗

arXiv2024

Ray class groups and ray class fields for orders of number fields

This paper contributes to the theory of orders of number fields. This paper defines a notion of "ray class group" associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. It shows existence of "ray class fields" corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. The paper gives exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes.

math.NT↗

arXiv2024

The family of $a$-floor quotient partial orders

An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced using the floor function. A positive integer $d$ is a floor quotient of $n$, denoted $d \,\preccurlyeq_{1}\, n$, if there is a positive integer $k$ such that $d = \lfloor{n / k}\rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies a family of partial orders, the $a$-floor quotient relations $\,\preccurlyeq_{a}\,$, for $a \in \mathbb{N}^+$, which interpolate between the floor quotient order and the divisor order on $\mathbb{N}^+$. The paper studies the internal structure of these orders.

math.NT↗

arXiv2024

The factorial function and generalizations, extended

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset $S$ of $\mathbb{Z}$. Bhargava's factorials $k!_S$ are invariants, constructed using the notion of $p$-orderings of $S$ where $p$ is a prime. This paper defines $b$-orderings of any nonempty subset $S$ of $\mathbb{Z}$ for all integers $b\ge2$, as well as "extreme" cases $b=1$ and $b=0$. It defines generalized factorials $k !_{S,T}$ and generalized binomial coefficients $\binom{k+\ell}{k}_{S,T}$ as nonnegative integers, for all nonempty $S$ and allowing only $b$ in $T\subseteq\mathbb{N}$. It computes $b$-ordering invariants when $S$ is $\mathbb{Z}$ and when $S$ is the set of all primes.

math.NT↗

arXiv2023

Asymptotics of Reciprocal Supernorm Partition Statistics

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers $p_i$ indexed by its parts $i$. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size $|λ|=n$, their perimeter equaling $n$, and their largest part equaling $n$. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to $e^γ \log n$ as $n \to \infty$.

math.CO↗

arXiv2021

The 3x+1 Problem: An Overview

This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the problem. It addresses two questions: (1) What can mathematics currently say about this problem? (as of 2010). (2) How can this problem be hard, when it is so easy to state?

math.NT↗

arXiv2021

Interleaving of path sets

Path sets are spaces of one-sided infinite symbol sequences corresponding to the one-sided infinite walks beginning at a fixed initial vertex in a directed labeled graph. Path sets are a generalization of one-sided sofic shifts. This paper studies decimation operations $ψ_{j, n}(\cdot)$ which extract symbol sequences in infinite arithmetic progressions (mod n). starting with the symbol at position j. It also studies a family of n-ary interleaving operations, one for each arity n, which act on an ordered set $(X_0, X_1, ..., X_{n-1})$ of one-sided symbol sequences on a finite alphabet A, to produce a set $X$ of all output sequences obtained by interleaving the symbols of words $x_i$ in each $X_i$ in arithmetic progressions (mod n). It studies a set of closure operations relating interleaving and decimation. It reviews basic algorithmic results on presentations of path sets and existence of a minimal right-resolving presentation. It gives an algorithm for computing presentations of decimations of path sets from presentations of path sets, showing the minimal right-resolving presentation of $ψ_{j,n}(X)$ has at most one more vertex than a minimal right-resolving presentation of X. It shows that a path set has only finitely many distinct decimations. It shows the class of path sets on a fixed alphabet is closed under all interleaving operations, and gives algorithms for computing presentations of n-fold interleavings of given sets $X_i$. It studies interleaving factorizations and classifies path sets that have infinite interleaving factorizations, and gives an algorithm to recognize them. It shows a finiteness of a process of iterated interleaving factorizations, which "freezes" factors that have infinite interleavings.

math.DS↗

arXiv2020

The Lerch zeta function and the Heisenberg group

This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch $L$-functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group $H^{J}$, called here the sub-Jacobi group, which is a semi-direct product of $GL(1, {\mathbb R})$ with the Heisenberg group $H({\mathbb R})$. The Heisenberg group action on L^2-functions on the Heisenberg nilmanifold $H({\mathbb Z}) \backslash H({\mathbb R})$ decomposes as $\bigoplus_{N \in {\mathbb Z}} H_N$, where each space $H_N~ (N \neq 0)$ consists of $|N|$ copies of an irreducible representation of $H({\mathbb R})$ with central character $e^{2 πi Nz}$. The paper shows that show one can further decompose $H_N (N \ne 0)$ into irreducible $H({\mathbb R})$-modules $H_{N,d}(χ)$ indexed by Dirichlet characters $(\bmod~ d)$ for $d \mid N$, each of which carries an irreducible $H^J$-action. On each $H_{N,d}(χ)$ there is an action of certain two-variable Hecke operators $\{T_m: m \ge 1\}$; these Hecke operators have a natural global definition on all of $L^2(H({\mathbb Z})\backslash H({\mathbb R}))$, including the space of one-dimensional representations $H_0$. For $H_{N,d}(χ)$ with $N \neq 0$ suitable Lerch $L$-functions on the critical line $\frac{1}{2} + it$ form a complete family of generalized eigenfunctions (pure continuous spectrum) for a certain linear partial differential operator $Δ_L$. These Lerch $L$-functions are also simultaneous eigenfunctions for all two-variable Hecke operators $T_m$ and their adjoints $T_m^{\ast}$, provided $(m, N/d) = 1$. Lerch $L$-functions are characterized by this Hecke eigenfunction property.

math.NT↗

arXiv2020

Decimation and Interleaving Operations in One-Sided Symbolic Dynamics

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show they are closed under composition. We also study a family of $n$-ary interleaving operations, one for each $n \ge 1$. Given subsets $X_0, X_1, ..., X_{n-1}$ of the shift space, the $n$-ary interleaving operator produces a set whose elements combine individual elements ${\bf x}_i$, one from each $X_i$, by interleaving their symbol sequences cyclically in arithmetic progressions $(\bmod\,n)$. We determine algebraic relations between decimation and interleaving operators and the shift operator. We study set-theoretic $n$-fold closure operations $X \mapsto X^{[n]}$, which interleave decimations of $X$ of modulus level $n$. A set is $n$-factorizable if $X=X^{[n]}$. The $n$-fold interleaving operators are closed under composition and are idempotent. To each $X$ we assign the set $\mathcal{N}(X)$ of all values $n \ge 1$ for which $X= X^{[n]}$. We characterize the possible sets $\mathcal{N}(X)$ as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. This class includes all shift-invariant sets. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable $X$, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.

math.DS↗

arXiv2020

Partial Factorizations of Products of Binomial Coefficients

Let $G_n= \prod_{k=0}^n \binom{n}{k},$ the product of the elements of the $n$-th row of Pascal's triangle. This paper studies the partial factorizations of $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having $p \le x$, counted with multiplicity. It shows $\log G(n, αn) \sim f_G(α)n^2$ as $n \to \infty$ for a limit function $f_{G}(α)$ defined for $0 \le α\le 1$. The main results are deduced from study of functions $A(n, x), B(n,x),$ that encode statistics of the base $p$ radix expansions of the integer $n$ (and smaller integers), where the base $p$ ranges over primes $p \le x$. Asymptotics of $A(n,x)$ and $B(n,x)$ are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.

math.NT↗

arXiv2020

Dilated floor functions having nonnegative commutator II. Negative dilations

This paper completes the classification of the set $S$ of all real parameter pairs $(α,β)$ such that the dilated floor functions $f_α(x) = \lfloor{αx}\rfloor$, $f_β(x) = \lfloor{βx}\rfloor$ have a nonnegative commutator, i.e. $ [ f_α, f_β](x) = \lfloor{α\lfloor{βx}\rfloor}\rfloor - \lfloor{β\lfloor{αx}\rfloor}\rfloor \geq 0$ for all real $x$. This paper treats the case where both dilation parameters $α, β$ are negative. This result is equivalent to classifying all positive $α, β$ satisfying $ \lfloor{α\lceil{βx}\rceil}\rfloor - \lfloor{β\lceil{αx}\rceil}\rfloor \geq 0$ for all real $x$. The classification analysis is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two generators.

math.NT↗

arXiv2020

Band-limited mimicry of point processes by point processes supported on a lattice

We say that one point process on the line $\mathbb{R}$ mimics another at a bandwidth $B$ if for each $n \ge 1$ the two point processes have $n$-level correlation functions that agree when integrated against all bandlimited test functions on bandwidth $[-B, B]$. This paper asks the question of for what values $a$ and $B$ can a given point process on the real line be mimicked at bandwidth $B$ by a point process supported on the lattice $a\mathbb{Z}$. For Poisson point processes we give a complete answer for allowed parameter ranges $(a,B)$, and for the sine process we give existence and nonexistence regions for parameter ranges. The results for the sine process have an application to the Alternative Hypothesis regarding the scaled spacing of zeros of the Riemann zeta function, given in a companion paper.

math.PR↗

arXiv2019

Higher Correlations and the Alternative Hypothesis

The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between nontrivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by T. Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$ which satisfy the Alternative Hypothesis spacing but mimic all statistics which are currently known about zeros of the zeta function.

math.NT↗

arXiv2018

Moser's Shadow Problem

Moser's shadow problem asks to estimate the shadow function $\mathfrak{s}_b(n)$, which is the largest number such that for each bounded convex polyhedron $P$ with $n$ vertices in $3$-space there is some direction ${\bf v}$ (depending on $P$) such that, when illuminated by parallel light rays from infinity in direction ${\bf v}$, the polyhedron casts a shadow having at least $\mathfrak{s}_b(n)$ vertices. A general version of the problem allows unbounded polyhedra as well, and has associated shadow function $\mathfrak{s}_u(n)$. This paper presents correct order of magnitude asymptotic bounds on these functions. The bounded case has answer $\mathfrak{s}_b(n) = Θ\big( \log (n)/ (\log(\log (n))\big$. The unbounded shadow problem is shown to have the different asymptotic growth rate $\mathfrak{s}_u(n) = Θ\big(1\big)$. Results on the bounded shadow problem follow from 1989 work of Chazelle, Edelsbrunner and Guibas on the (bounded) silhouette span number $\mathfrak{s}_b^{\ast}(n)$, defined analogously but with arbitrary light sources. We complete the picture by showing that the unbounded silhouette span number $\mathfrak{s}_u^{\ast}(n)$ grows as $Θ\big( \log (n)/ (\log(\log (n))\big)$.

math.MG↗

arXiv2018

Dilated floor functions having nonnegative commutator I. Positive and mixed sign dilations

In this paper and its sequel we classify the set $S$ of all real parameter pairs $(α,β)$ such that the dilated floor functions $f_α(x) = \lfloor{αx}\rfloor$ and $f_β(x) = \lfloor{βx}\rfloor$ have a nonnegative commutator, i.e. $ [ f_α, f_β](x) = \lfloor{α\lfloor{βx}\rfloor}\rfloor - \lfloor{β\lfloor{αx}\rfloor}\rfloor \geq 0$ for all real $x$. The relation $[f_α,f_β]\geq 0$ induces a preorder on the set of non-zero dilation factors $α, β$, which extends the divisibility partial order on positive integers. This paper treats the cases where at least one of the dilation parameters $α$ or $β$ is nonnegative. The analysis of the positive dilations case is related to the theory of Beatty sequences and to the Diophantine Frobenius problem in two generators.

math.NT↗

arXiv2018

Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem

The problem of twelve spheres is to understand, as a function of $r \in (0,r_{max}(12)]$, the configuration space of $12$ non-overlapping equal spheres of radius $r$ touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of $N$ spheres of radius $r$ touching a central unit sphere, with emphasis on $3 \le N \le 14$. The problem of determining the maximal radius $r_{max}(N)$ is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.

math.MG↗