Papers by Jeffrey Lagarias
Discrete and Computational Geometry, Jan 22, 2003
Substitution Delone set families are families of Delone sets X = (X 1 ,. .. , X n) which satisfy ... more Substitution Delone set families are families of Delone sets X = (X 1 ,. .. , X n) which satisfy the inflation functional equation X i = m j=1 (A(X j) + D ij), 1 ≤ i ≤ m, in which A is an expanding matrix, i.e., all of the eigenvalues of A fall outside the unit circle. Here the D ij are finite sets of vectors in R d and denotes union that counts multiplicity. This paper characterizes families X = (X 1 ,. .. , X n) that satisfy an inflation functional equation, in which each X i is a multiset (set with multiplicity) whose underlying set is discrete. It then studies the subclass of Delone set solutions, and gives necessary conditions on the coefficients of the inflation functional equation for such solutions X to exist. It relates Delone set solutions to a narrower subclass of solutions, called self-replicating multi-tiling sets, which arise as tiling sets for self-replicating multi-tilings.
Springer eBooks, Jul 18, 2007
ABSTRACT We describe connections between the de Branges theory of Hilbert spaces of entire functi... more ABSTRACT We describe connections between the de Branges theory of Hilbert spaces of entire functions and the Riemann hypothesis for Dirichlet L-functions. Assuming the Riemann hypothesis holds for a given L-function, there exists an associated de Branges space with interesting properties, and conversely. This de Branges space comes with an associated self-adjoint operator having as eigenvalues the imaginary parts of the L-function zeros on the critical line, and this operator has an interpretation as a “Hilbert-Polya” generalized differential operator.
Ergodic Theory and Dynamical Systems, Apr 1, 1997
Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=... more Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.
Canadian Mathematical Bulletin
We improve upon the traditional error term in the truncated Perron formula for the logarithm of a... more We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.
Ergodic Theory and Dynamical Systems, Feb 1, 1997
Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=... more Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic on the unit disk $|z|<1$ and analytic on $|z|<1/\beta$. This paper shows that the singularities of $L_{\beta,\alpha}(z)$ on the circle $|z|=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N):0\le l\le N-1\}$, for some integer $N\ge 1$. Here $N$ can be taken to be the period $N_{\beta,\alpha}$ of a certain Markov chain $\Sigma_{\beta,\alpha}$ which encodes information about generalized lap numbers $L_{n}(i,j)$ of $f_{\beta,\alpha}$, where $L_{n}(i,j)$ counts monotonic pieces of $f_{\beta,\alpha}^{n}$ whose image is $[f^{i}(0),f^{j}(1^{-}))$. We show that $N_{\beta,\alpha}=1$ whenever $\beta>2$. Finally, we give the criterion that $N_{\beta,\alpha}=1$ if and only if for all $n\ge 1$ the map $f_{\beta,\alpha}^{n}$ is ergodic with respect to the maximal entropy measure of $f_{\beta,\alpha}$.
Ergodic Theory and Dynamical Systems, Apr 1, 1997
Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=... more Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.
arXiv (Cornell University), Nov 25, 2015
This paper gives a representation-theoretic interpretation of the Lerch zeta function and related... more 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, R) with the Heisenberg group H(R). The Heisenberg group action on L 2-functions on the Heisenberg nilmanifold H(Z)\H(R) decomposes as N∈Z HN , where each space HN (N = 0) consists of |N | copies of an irreducible infinite-dimensional representation of H(R) with central character e 2πiNz. The paper shows that show one can further decompose HN (N = 0) into irreducible H(R)-modules H N,d (χ) indexed by Dirichlet characters (mod d) for d | 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 {Tm : m ≥ 1}; these Hecke operators have a natural global definition on all of L 2 (H(Z)\H(R)), including the space of one-dimensional representations H0. For H N,d (χ) with N = 0 suitable Lerch L-functions on the critical line 1 2 + it form a complete family of generalized eigenfunctions (purely continuous spectrum) for a certain linear partial differential operator ∆L. These Lerch L-functions are also simultaneous eigenfunctions for all two-variable Hecke operators Tm and their adjoints T * m , provided (m, N/d) = 1. Lerch L-functions are characterized by this Hecke eigenfunction property.
arXiv (Cornell University), Nov 25, 2015
This paper gives a representation-theoretic interpretation of the Lerch zeta function and related... more 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, R) with the Heisenberg group H(R). The Heisenberg group action on L 2-functions on the Heisenberg nilmanifold H(Z)\H(R) decomposes as N∈Z HN , where each space HN (N = 0) consists of |N | copies of an irreducible infinite-dimensional representation of H(R) with central character e 2πiNz. The paper shows that show one can further decompose HN (N = 0) into irreducible H(R)-modules H N,d (χ) indexed by Dirichlet characters (mod d) for d | 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 {Tm : m ≥ 1}; these Hecke operators have a natural global definition on all of L 2 (H(Z)\H(R)), including the space of one-dimensional representations H0. For H N,d (χ) with N = 0 suitable Lerch L-functions on the critical line 1 2 + it form a complete family of generalized eigenfunctions (purely continuous spectrum) for a certain linear partial differential operator ∆L. These Lerch L-functions are also simultaneous eigenfunctions for all two-variable Hecke operators Tm and their adjoints T * m , provided (m, N/d) = 1. Lerch L-functions are characterized by this Hecke eigenfunction property.
Notices of the American Mathematical Society
Mathematicians always queued to hear John Conway speak and delved into his writing. They expected... more Mathematicians always queued to hear John Conway speak and delved into his writing. They expected to be entertained by beautiful mathematics and believed that they would emerge with valuable enlightenment. John welcomed this attention and considered it his duty to make his mathematics elegant. What really made his mathematics valuable was his wealth of insight. Wherever he worked, he opened up avenues for us to follow. John challenged and inspired us in many different ways. His Game of Life has been investigated by thousands, while his river method provides a novel approach to the classical subject of quadratic forms. His orbifold notation was devised in collaboration with Bill Thurston. It gives a new way to describe symmetry patterns like those in the opening images above. The images are taken from the beautiful book, The Symmetries of Things, coauthored by John, Heidi Burgiel, and Chaim Goodman-Strauss. Group theorists, like me, are drawn to his paper "Monstrous Moonshine," written with his friend Simon Norton. John and Simon filled the paper with what were outlandish examples and provocative conjectures about the
arXiv (Cornell University), Nov 7, 2004
The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by ... more The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2k+1 3k+2 : k ≥ 0} together with {2}. This semigroup encodes backwards iteration under the 3x + 1 map, and the 3x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b ≡ 0(mod 3), and so contains all positive integers.
2019-20 MATRIX Annals, 2021
The American Mathematical Monthly, 2016
Bulletin of the American Mathematical Society, 2015
arXiv (Cornell University), Jun 3, 2022
We improve upon the traditional error term in the truncated Perron formula for the logarithm of a... more We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tang... more Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain, where a Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Part I showed there exists a discrete group, the Apollonian group, acting on a parameter space of (ordered, oriented) Descartes configurations, such that the Descartes configurations in a packing formed an orbit under the action of this group. It is observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested * Ronald L.
arXiv (Cornell University), Jan 7, 2021
Path sets are spaces of one-sided infinite symbol sequences corresponding to the one-sided infini... more 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 G. Path sets are a generalization of one-sided sofic shifts. This paper studies decimation operations ψj,n(•) 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 ⊛n, one for each n ≥ 1, which act on an ordered set (X0, X1, ..., Xn−1) of one-sided symbol sequences X0⊛X1⊛ • • • ⊛Xn−1 on an alphabet A, by interleaving the symbols of each Xi in arithmetic progressions (mod n), It studies a set of closure operations relating interleaving and decimation. This paper gives 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 ψj,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 interleavings and gives an algorithm to compute presentations of interleavings of path sets. It studies interleaving factorizations and classifies path sets that have infinite interleaving factorizations and gives an algorithm to recognize them. It shows the finiteness of a process of iterated interleaving factorizations, which "freezes" factors that have infinite interleavings. CONTENTS WILLIAM C. ABRAM, JEFFREY C. LAGARIAS, AND DANIEL SLONIM 5.2. Interleaving closures of path sets 21 6. Interleaving factorizations 21 6.1. Structure of interleaving factors:arbitrary sets X 21 6.2. Infinitely factorizable path sets 22 6.3. Size of minimal right-resolving presentations for interleaving factors of path sets 23 7. Structure of Infinitely Factorizable Path Sets 23 7.1. Characterization of infinitely factorizable path sets 23 7.2. Bounds for the number of distinct factors of infinitely factorizable path sets 25 7.3. Minimal right-resolving presentations for interleaving factors-part 2 26 8. Finitely Factorizable Path Sets 27 8.1. Bounds for number of distinct n-fold interleaving factorizations 27 8.2. Iterated interleaving and complete factorizations 29 9. Concluding Remarks 31 9.1. Automatic sequences associated to path sets 31 Appendix A. Path sets in Automata Theory 31 A.1. Automata and languages 31 A.2. Automata-theoretic characterization of path sets 32 A.3. Minimal deterministic automata 33 A.4. Path set languages and closures of rational languages 33 Appendix B. Self-interleaving criterion 34 References 35
Ergodic Theory and Dynamical Systems, 1996
Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1)... more Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of monotonic pieces of the nth iterate . We derive an explicit factorization formula for Lβα(z) which directly shows that Lβα(z) is a function meromorphic in the open unit disk {z: |z| < 1} and analytic in the open disk {z: |z| < 1/β}, with a simple pole at z = 1/β.Comparison with a known formula for the Artin—Mazur—Ruelle zeta function ζβ,α(z) of fβα shows that Lβα(z) and ζβ,α(z) have identical sets of singularities in the disk {z: |z| < 1}. We derive two more factorization formulae for Lβ,α(z) valid for certain parameter ranges of (β, α). When 1 < α + β ≤ 2, there is sometimes a ‘renormalization’ structure of such maps present, which has previously been studied in connection with simplified models for the Lorenz attractor. In the case that fβα is...
We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is u... more We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
arXiv (Cornell University), Sep 11, 2000
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent c... more Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: x 2 + y 2 + z 2 + w 2 = 1 2 (x + y + z + w) 2. Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element −n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. We present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions. Finally, we discuss asymptotic properties of the set of curvatures obtained as the packing is recursively constructed from a root quadruple.
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Papers by Jeffrey Lagarias