Colourings of quasicrystals

1994 ◽  
Vol 72 (7-8) ◽  
pp. 442-452 ◽  
Author(s):  
R. V. Moody ◽  
J. Patera
Keyword(s):  
Number Field ◽  
Penrose Tiling ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Arithmetic Properties ◽  
Weight Lattice

We introduce a notion of colouring the points of a quasicrystal analogous to the idea of colouring or grading of the points of a lattice. Our results apply to quasicrystals that can be coordinatized by the ring R of integers of the quadratic number field [Formula: see text] and provide a useful and wide ranging tool for determining of sub-quasicrystals of quasicrystals. Using the arithmetic properties of R we determine all possible finite colourings. As examples we discuss the 4-colours of vertices of a Penrose tiling arising as a subset of 5-colouring of an R lattice, and the 4-colouring of quasicrystals arising from the D6 weight lattice.

scholarly journals Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

scholarly journals Geometric-progression-free sets over quadratic number fields

2017 ◽  
Vol 147 (2) ◽  
pp. 245-262
Author(s):  
Andrew Best ◽  
Karen Huan ◽  
Nathan McNew ◽  
Steven J. Miller ◽  
Jasmine Powell ◽  
...  
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Geometric Progression ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Quadratic Number Fields ◽  
Geometric Progressions ◽  
Upper Density ◽  
Free Sets

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

A SYSTEM OF RELATIVE PELLIAN EQUATIONS AND A RELATED FAMILY OF RELATIVE THUE EQUATIONS

2006 ◽  
Vol 02 (04) ◽  
pp. 569-590 ◽  
Author(s):  
BORKA JADRIJEVIĆ ◽  
VOLKER ZIEGLER
Keyword(s):  
Number Field ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Root Of Unity ◽  
Related Family ◽  
The Family ◽  
Thue Equation ◽  
Imaginary Quadratic Number ◽  
Pellian Equations

In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field [Formula: see text]. We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation [Formula: see text] and solve it by the method of Tzanakis under the same assumptions.

scholarly journals Picard Groups and Refined Discrete Logarithms

2005 ◽  
Vol 8 ◽  
pp. 1-16 ◽  
Author(s):  
W. Bley ◽  
M. Endres
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Finite Abelian Group ◽  
Discrete Logarithm ◽  
Quadratic Number ◽  
Prime Degree ◽  
Quadratic Number Field ◽  
Picard Groups ◽  
Imaginary Quadratic Number ◽  
Ring Of Algebraic Integers

AbstractLet K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

scholarly journals On a criterion for the class number of a quadratic number field to be one

1980 ◽  
Vol 79 ◽  
pp. 123-129 ◽  
Author(s):  
Masakazu Kutsuna
Keyword(s):  
Number Field ◽  
Class Number ◽  
Euclidean Algorithm ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Imaginary Quadratic Number ◽  
Necessary And Sufficient

G. Rabinowitsch [3] generalized the concept of the Euclidean algorithm and proved a theorem on a criterion in order that the class number of an imaginary quadratic number field is equal to one:Theorem.It is necessary and sufficient for the class number of an imaginary quadratic number fieldD= 1 — 4m, m> 0,to be one that x2—x+m is prime for any integer x such that1 ≤x≤m— 2.

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