Geodesic
Beyond Expansion IV: Traces of Thin Semigroups
Discrete analysis (2018) Cet article a éte moissonné depuis la source Scholastica

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We continue 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. 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 alpha > 0, we are able to produce the exponent alpha 1/3. 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).
Publié le : 
@article{DAS_2018_a15,
     author = {Jean Bourgain and Alex Kontorovich},
     title = {Beyond {Expansion} {IV:} {Traces} of {Thin} {Semigroups}},
     journal = {Discrete analysis},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2018_a15/}
}
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AU  - Alex Kontorovich
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JO  - Discrete analysis
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UR  - http://geodesic.mathdoc.fr/item/DAS_2018_a15/
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%A Jean Bourgain
%A Alex Kontorovich
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Jean Bourgain; Alex Kontorovich. Beyond Expansion IV: Traces of Thin Semigroups. Discrete analysis (2018). http://geodesic.mathdoc.fr/item/DAS_2018_a15/