On Representations of Integers in Thin Subgroups of SL2(ℤ)

Jean Bourgain, Alex Kontorovich

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26 Scopus citations


Let be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v0,w0 ∈ ℤ2. We consider the set S of all integers occurring in 〈 v0 γ,w0 〉, for γ ∈ and the usual inner product on ℝ. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd's 5/6-th spectral gap in infinite-volume, we show that S contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set E(N) of integers {pipe}n{pipe} < N which are locally admissible (n ∈ S(mod q) for all q ≥ 1) but fail to be globally represented, n ∉ S, has a power savings, {pipe}E (N){pipe} ≪ N1 - 0 for some 0 > 0, as N → ∞

Original languageEnglish (US)
Pages (from-to)1144-1174
Number of pages31
JournalGeometric and Functional Analysis
Issue number5
StatePublished - Nov 2010
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology


  • Fuchsian groups of the second kind
  • circle method
  • exponential sums
  • local-global principle


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