Sublattices of Finite Index
نویسنده
چکیده
Assuming the Gowers Inverse conjecture and the Möbius conjecture for the finite parameter s, Green-Tao verified Dickson’s conjecture for lattices which are ranges of linear maps of complexity at most s. In this paper, we reformulate GreenTao’s theorem on Dickson’s conjecture, and prove that, if L is the range of a linear map of complexity s, and L1 is a sublattice of L of finite index, then L1 is the range of a linear map of complexity s.
منابع مشابه
Some algebraic properties of crystallographic sublattices.
In this article, a number of the results relevant to the concept of sublattices of a basic crystallographic lattice are reviewed, emphasizing particularly previously unpublished work on the algebraic aspects. A three-dimensional geometric lattice L can be considered as an infinite Abelian group under addition. A sublattice S of L, which is also three-dimensional, is a subgroup of L such that th...
متن کاملSublattices of Certain Coxeter Lattices
In this paper, we describe the sublattices of some lattices, extending previous results of [Ber]. Our description makes intensive use of graphs. Résumé. Dans cet article, nous décrivons les sous-réseaux de certains réseaux de Coxeter, prolongeant les résultats de [Ber]. Notre description utilise des graphes. In his 1951 paper Extreme forms ([Cox]), the late Coxeter studied lattices Λ which sati...
متن کاملSimilar Sublattices of the Root Lattice a 4
Similar sublattices of the root lattice A4 are possible [9] for each index that is the square of a non-zero integer of the form m + mn − n. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q( √ 5)) and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given ...
متن کاملSimilar Sublattices and Coincidence Rotations of the Root Lattice A4 and Its Dual
A natural way to describe the Penrose tiling employs the projection method on the basis of the root lattice A4 or its dual. Properties of these lattices are thus related to properties of the Penrose tiling. Moreover, the root lattice A4 appears in various other contexts such as sphere packings, efficient coding schemes and lattice quantizers. Here, the lattice A4 is considered within the icosia...
متن کاملProjectional Entropy and the Electrical Wire Shift – Preliminary Version
In this paper we present an extendible, block gluing Z shift of finite type W el in which the topological entropy equals the L-projectional entropy for a two-dimensional sublattice L ( Z, even so W el is not a full Z extension of W el L . In particular this example shows that Theorem 4.1 of [3] does not generalize to r-dimensional sublattices L for r > 1. Nevertheless we are able to reprove and...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007