Automated Vehicles (AV) promise many benefits for future mobility. One of them is a reduction of the required total vehicle fleet size, especially if AVs are used predominantly as shared vehicles. This paper presents research on this potential reduction for the greater Zurich region, Switzerland. Fleets of shared AVs, serving a predefined demand, are simulated with a simulation framework introduced in the paper. Different scenarios are created, combining different levels of demand for AVs with different levels of supply (i.e. AV fleet size). An important contribution of this study is the use of a spatially and temporally highly detailed travel demand, going beyond the simplifications of previous studies on the topic. This provides a more solid basis to the ongoing discussion on the future fleet size. It is found that, for a given fleet performance target (here 95% of all transport requests are served within 5 minutes), the relationship between served demand and required fleet size is non-linear and the ratio increases as demand increases. There is a scale effect, which has the important implication that for different levels of demand the fleet is used more or less efficiently. This paper also finds that, if waiting times of up to 10 minutes are accepted, a reduction of up to 90% of the total vehicle fleet can be possible even without active fleet management like vehicle redistribution. Such effects require, however, that a large enough share of the car demand can be served by AVs.