The Principle of the Large Sieve

We describe a very general abstract form of sieve based on a large-sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We give some first results on the number of prime divisors of “most” elements of an elliptic divisibility sequence, and we develop in some detail “probabilistic” sieves for random walks on arithmetic groups, e.g., estimating the probability of finding a reducible characteristic polynomial at some step of a random walk on SL(n,Z). In addition to the sieve principle, the applications depend on bounds for a large sieve constant. To prove such bounds involves a variety of deep results, including Property (τ ) or expanding properties of Cayley graphs, and the Riemann Hypothesis over finite fields.