2006/09/30 by Emmanuel Kowalski, Kowalski, Emmanuel
Mathematics · #11B37 #11C99 #11N35 #11N36 #14G15 #20C33 #22D10 #60G50 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Probability (math.PR) #math.GR #math.NT #math.PR #msc:11B37 #msc:11C99 #msc:11N35 #msc:11N36 #msc:14G15 #msc:20C33 #msc:22D10 #msc:60G50
paper · pdf · doi:10.48550/arxiv.math/0610021
65 pages; some small corrections, and added application to random walk on mapping class groups (non pseudo-Anosov elements form a transient set), based on work of Maher and Rivin
arxiv created 2006/10/29 · arxiv updated 2009/12/01
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 get 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 (T) or expanding properties of Cayley graphs, and the Riemann Hypothesis over finite fields. It seems likely that this sieve can have further applications.