This paper constitutes Part IV in our study of particular instances of the Affine Sieve, producing levels of distribution beyond those attainable from expansion alone. Motivated by McMullen's Arithmetic Chaos Conjecture regarding low-lying closed geodesics on the modular surface defined over a given number field, we study the set of traces for certain sub-semi-groups of SL2(ℤ) corresponding to absolutely Diophantine numbers (see §1.2). In particular, we are concerned with the level of distribution for this set. While the standard Affine Sieve procedure, combined with Bourgain-Gamburd-Sarnak's resonance-free region for the resolvent of a "congruence" transfer operator, produces some exponent of distribution α > 0, we are able to produce the exponent α = 1=3-ε. This recovers unconditionally the same exponent as what one would obtain under a Ramanujan-type conjecture for thin groups. A key ingredient, of independent interest, is a bound on the additive energy of SL2(ℤ).
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Additive energy
- Affine sieve
- Thin groups