The Ergodic Theory of Lattice Subgroups

By Alexander Gorodnik, A. Nevo. 2006

124 citations recorded by the source corpus.

We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quantitative mean ergodic theorem for a family of averages gives rise to a quantitative solution of the lattice point counting problem in their supports. We demonstrate the new explicit error terms that we obtain by a variety of examples.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66223f2edffc0fce6f87794457e1e2870ec2a4b2

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In Mathematical Physics

1. Peter Sarnak
2. Alexander Gorodnik
3. A. Nevo
4. A. Mohammadi
5. H. Oh
6. V. Blomer
7. Akshay Venkatesh
8. Gergely Harcos

Citations in the corpus

• Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “, when δ = n− 1, both the effective counting and applications to an affine sieve have been extensively studied (see [16], [17], [4], [46], [21], [42],[48], [20], etc.”
• Quantitative ergodic theorems and their number-theoretic applications (2013, methodology): “For more general formulations, we refer to [GN10].”
• Prime and almost prime integral points on principal homogeneous spaces (2009, methodology): “We now turn to the proof of Theorem 3.2, and begin by explicating the connection between the averaging operators associated with t and counting lattice points, following the method developed in [ GN1 ].”
• Counting lattice points (2009, background): “In addition to numerous examples discussed in [32], Chapter 2, let us mention very 135 Gorodnik and Nevo, Counting lattice points”