Posted by Jack on February 14, 2014, 4:26 pm, in reply to "Re: Definiton of Lattice"
I was a bit worried about whether this condition was indeed sufficient, but thanks to the helpful people on Math StackExchange here:
http://math.stackexchange.com/questions/672654/lattice-definition-as-finite-density-infinite-set-of-vectors-closed-under-additi
I can confirm that an infinite set of vectors closed under subtraction with a finite density of points, or equivalently without an arbitrarily small vector in the set, is indeed equivalent to the integer sum definition of the lattice.
I'd post the full proof from Siegel's Lectures on the Geometry of Numbers here but it's rather long. Basically it proceeds by formally constructing a basis for the set with the smallest possible vectors, then showing that if there is a vector which cannot be written as an integer sum of this basis one could construct an even smaller basis, in contradiction to the original assumption.
45
Message Thread
« Back to index