arXiv subjects

Jeffrey C. Lagarias

Publications and source records attributed to Jeffrey C. Lagarias.

80 records · Page 5Linked to original sources

arXiv2001

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2,Z) with respect to the standard fundamental domain F = {z=x+iy: -1/2 <= x <= 1/2 and |z|>=1} of PSL(2,Z). The cutting sequence for a vertical geodesic {θ+it: t > 0} is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space {L,R,J} which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain F up to an isometry of the hyperbolic plane H. We give conversion methods between the cutting sequence for the vertical geodesic {θ+it: t > 0}, the MGCF expansion of θand the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θcan be computed from the cutting sequence for the vertical geodesic θ+it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first l symbols when given as input the first O(l) symbols of the ACF expansion, which takes time O(l^2) and space O(l).

math.DS↗

arXiv2001

Lower bounds for the total stopping time of 3X+1 iterates

The 3X+1 function T(n) is (3n+1)/2 if n is odd and n/2 if n is even. The total stopping time σ_\infty (n) for a positive integer n is the number of iterations of the 3x+1 function to reach 1 starting from n, and is \infty if 1 is never reached. The 3x+1 conjecture states that this function is finite. We show that infinitely many n have a finite total stopping time with σ_\infty(n) > 6.14316 log n. The proof uses a very large computation. It is believed that almost all positive integers have σ_\infty (n) > 6.95212 \log n. The method of the paper should extend to prove infinitely many integers have this property, but it would require a much larger computation.

math.NT↗

arXiv2001

Beyond the Descartes circle theorem

The Descartes circle theorem states that if four circles are mutually tangent with disjoint intersion, then their curvatures (or "bends) b_j = 1/r_j satisfy the relation (b_1 + b_2 + b_3 + b_4)^2 = 2(b_1^2 + b_2^2 + b_3^2 + b_4^2). We show that similar relations hold involving the centers of the circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes theorem. These relations have matrix generalizations to the n-dimensional case, in each of Euclidean, spherical and hyperbolic geometries, and they include a Descartes circle theorem for spherical and hyperbolic space.

math.MG↗

arXiv2000

Bounds for Local Density of Sphere Packings and the Kepler Conjecture

This paper describes the local density inequality approach to getting upper bounds for sphere packing densities in R^n. This approach was first suggested by L. Fejes-Toth in 1956 to prove the Kepler conjecture that the densest sphere packing in R^3 is the "cannonball packing". The approaches of L. Fejes-Toth, W.-Y. Hsiang, and T. Hales to the Kepler conjecture are each based on (different) local density inequalities. Recently T. Hales and S. P. Ferguson have presented extensive details carrying out a modified version of the otrignal Hales approach to prove the Kepler conjecture. We describe the particular local density inequality undelying the Hales and Ferguson approach and sketch some features of their proof.

math.MG↗

arXiv2000

Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups

A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X) has an orthogonal basis of exponentials. It is conjectured that every spectral set tiles R^n by translations. A set of translations T has a universal spectrum if every set that that tiles by translations by T has this spectrum. A recent result proved that many periodic tiling sets have universal spectra, using results from factorizations of abelian groups, for groups for which a strong form of a conjecture of Tijdeman is valid. This paper shows Tijdeman's conjecture does not hold for the cyclic group of order 900. It formulates a new sufficient conjecture for a periodic tiling set to have a universal spectrum, and uses it to show that the tiling sets for the counterexample above do have universal spectra.

math.FA↗

arXiv1998

The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc 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 {\bf PSPACE}. We also give exponential worst-case 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.

math.GT↗

arXiv1998

The number of Reidemeister Moves Needed for Unknotting

There is a positive constant $c_1$ such that for any diagram $D$ representing the unknot, there is a sequence of at most $2^{c_1 n}$ Reidemeister moves that will convert it to a trivial knot diagram, $n$ is the number of crossings in $D$. A similar result holds for elementary moves on a polygonal knot $K$ embedded in the 1-skeleton of the interior of a compact, orientable, triangulated $PL$ 3-manifold $M$. There is a positive constant $c_2$ such that for each $t \geq 1$, if $M$ consists of $t$ tetrahedra, and $K$ is unknotted, then there is a sequence of at most $2^{c_2 t}$ elementary moves in $M$ which transforms $K$ to a triangle contained inside one tetrahedron of $M$. We obtain explicit values for $c_1$ and $c_2$.

math.GT↗

arXiv1992

Keller's cube-tiling conjecture is false in high dimensions

O. H. Keller conjectured in 1930 that in any tiling of $\Bbb R^n$ by unit $n$-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $n\le 6$. We show that for all $n\ge 10$ there exists a tiling of $\Bbb R^n$ by unit $n$-cubes such that no two $n$-cubes have a complete facet in common.

math.MG↗