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MiS Preprint
50/2001
Embeddings of lattices in L^2([0,1], Z)
Valerii N. Berestovskii and Conrad Plaut
Abstract
We show how to construct the group $L^2 ([0,1], \mathbb{Z})$ using any sequence of Hadamard matrices. This construction is nicely compatible with the classical Haar and Rademacher functions. We then show that every k-dimensional Euclidean lattice is isometrically isomorphic to a k-slice of $L^2 ([0,1], \mathbb{Z})$. Finally we prove a similar embedding theorem for integral and p-rational lattices into the $\mathbb{Z}$-module of all continuous integer-valued functions on the group $\mathbb{Z}_p$ of p-adic integers.
