Transference principles and locally symmetric spaces

Abstract

We explain how the Transference Principles from Diophantine approximation can be interpreted in terms of geometry of the locally symmetric spaces $T_n=SO(n) \backslash SL(n,R) /SL(n,Z)$ with $n>1$, and how, via this dictionary, they become transparent geometric remarks and can be easily proved. Indeed, a finite family of linear forms is naturally identified to a locally geodesic ray in a space $T_n$ and the way this family is approximated is reflected by the heights at which the ray rises in the cuspidal end. The only difference between the two types of approximation appearing in a Transference Theorem is that the height is measured with respect to different rays in $W$, a Weyl chamber in $T_n$. Thus the Transference Theorem is equivalent to a relation between the Busemann functions of two rays in $W$. This relation is easy to establish on $W$, because restricted to it the two Busemann functions become two linear forms. Since $T_n$ is at finite Hausdorff distance from $W$, the same relation is satisfied up to a bounded perturbation on the whole of $T_n$.