Manfred Einsiedler, ETH Z"urich

Abstract:

In joint work with Shahar Mozes we derive results of the following type: "Given a primitive integer quaternion a very few other integer quaternions are not divisible by a." In the proof these results depend on measure rigidity of higher rank Cartan actions respectively adelic equidistribution of adelic torus actions. Of course, care must be taken in formulation of the result as there are actually many integer quaternions that are not divisible by a. Here are the two main theorems:

Given a as above there exists some rational integer A such that every integer quaternion whose norm is divisible by A is divisible by a.

Given two split primes p and q, define the sub semi group of quaternions whose norm is a product of powers of p and q. Then for any primitive element a in this semi group, there exists some rational integer A such that the set of integer quaternions in the semi group that are not divisible by a but whose norm is divisible by A is of sub-exponential growth.