Embedding lattices in $L^{2}([0,1]\mathbb{Z})$

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 n-dimensional Euclidean lattice is isometrically isomorphic to a n-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.