Counting and dynamics in SL_2
Advances in Homogeneous Dynamics May 11, 2015 - May 15, 2015
Location: MSRI: Simons Auditorium
counting points on a lattice
11K55 - Metric theory of other algorithms and expansions; measure and Hausdorff dimension [See also 11N99, 28Dxx]
11K50 - Metric theory of continued fractions [See also 11A55, 11J70]
11Kxx - Probabilistic theory: distribution modulo $1$; metric theory of algorithms
37-XX - Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
In this talk I'll discuss a lattice point count for a thin semigroup inside SL_2(Z). It is important for applications I'll describe that one can perform this count uniformly throughout congruence classes.
The approach to counting is dynamical - with input from both the real place and finite primes. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play. At finite places the necessary expansion property
follows from work of Bourgain and Gamburd (at one prime) or Bourgain, Gamburd and Sarnak (at squarefree moduli).
These are underpinned by tripling estimates in SL_2(F_p) due to Helfgott. I'll try to explain in simple terms the key dynamical facts behind all of these methods.
This talk is based on joint work with Hee Oh and Dale Winter.
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