Automorphs

Abstract

A homogeneous linear transformation ω is said to be an automorph of a point set L if L is just the set of points ωx,x∈L. The automorphs of a set L evidently form a group. Many of the point sets of interest in the geometry of numbers, or which occur naturally in problems arising in other branches of number-theory, have a rich group of automorphs which is reflected in the set of L-admissible lattices. Already in the work in which he introduced the notion of limit of a sequence of lattices, MAHLER (1946d, e) laid the foundations for future work and indicated some fundamental theorems. Since then much has been done but some challenging and natural questions remain unanswered.