M 1011

Similar Sublattices of Higher Dimensional Lattices Rudolf Scharlau Technische Universität Dortmund Abstract. The similar sublattices (SSLs) of a given lattice in Euclidean d-space are by definition the images of the lattice under similarities (norm-stretching maps). As objects of discrete geometry, they are of interest in their own right, and they have also shown up in various applications: crystallography, material science, digital coding, more precisely ``multiple description`` vector quantizers. In the 2-dimensional case, a rather complete picture of all SSLs has recently been given by Baake, Zeiner and the speaker. Less complete results exist in dimensions 3 and 4. In higher dimensions, only partial results on the existence of SSLs have been published by Conway, Rains and Sloane. In this talk, I propose a systematic approach to SSLs of integral lattices of (in principle) arbitrary dimension, based on the arithmetic theory of integral quadratic forms. In ``good`` cases, this not only answers the existence question, but also leads to a precise counting of the number of SSLs of a given index.

Rudolf Scharlau

Dortmund