NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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<strong>IN</strong>HOMOGENEOUS QUADRATIC FORMS 127 Let the Hamiltonian be the geodesic flow for flat 2-tori. Sarnak [Sar] proved that, for almost all 2-dimensional flat tori (with respect to Lebesgue measure on the moduli space of 2-dimensional flat tori), the pair correlation density function converges to the pair correlation density of a Poisson process. He used averaging arguments to reduce the pair correlation problem to one of spacing between the values at integers of binary quadratic forms. This is related to the quantitative Oppenheim problem in the case of signature (2, 2). One corollary of Theorem 1.3 is that the Berry-Tabor conjecture holds for the pair correlation of 2-dimensional flat tori under certain explicit Diophantine conditions. Similarly, Theorem 1.9 has a corollary in this direction. Let h be a lattice in R 2 . Also, let α = (α1,α2) ∈ R 2 . Now the eigenvalues of the Laplacian − =− ∂2 ∂2 − ∂y2 ∂y2 with quasi-periodicity conditions (17) φ(x + v) = e 2πi〈α,v〉 φ(x) for all x ∈ R 2 and all v ∈ h (18) are of the form 4π 2 w + α 2 , where w ∈ h ∗ and where h ∗ is the dual lattice to h. Let 0 ≤ λ0
128 MARGULIS and MOHAMMADI α ∈ R 2 be given, and define β as above. Suppose that at least one of the following holds. (i) The vector β = (β1,β2) is Diophantine. (ii) There exists N,C > 0 such that, for all triples of integers (p1,p2,q) with q ≥ 2, we have max Ai − i=1,2 pi > q C q Then for any interval (a,b) with 0 /∈ (a,b), we have lim T →∞ Rh,α(a,b,T ) = c 2 h (b − a). (22) Hence the spectrum satisfies the Berry-Tabor conjecture for the pair correlation function. The case of h = Z 2 was proved by Marklof [Mark]. His approach utilizes results from the theory of unipotent flows combined with an application of theta sums. We also use the theory of unipotent flows in our proof; however, our strategy to control the integral of unbounded functions over certain orbits is dynamical and rests heavily on [EMM1] and [EMM2]. Outline of the proof. Let Qξ be a quadratic form of signature (p, q),andletn = p+q. Fix an interval (a,b), andletU⊂Rnbe a “suitably chosen" compact set such that a
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NEAR OPTIMAL BOUNDS IN FREIMAN’S
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SUR LA NON-DENSITÉ DES POINTS ENTI
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30 CASIM ABBAS Recall that the Reeb
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32 CASIM ABBAS that Here the energy
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34 CASIM ABBAS • uτ ( ˙S) ∩ u
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36 CASIM ABBAS punctured surfaces w
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38 CASIM ABBAS that is, the central
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40 CASIM ABBAS even have A(r) ≡ 0
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42 CASIM ABBAS are contact forms on
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44 CASIM ABBAS the standard complex
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46 CASIM ABBAS and Jδε2 = xγ1(r)
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48 CASIM ABBAS (λ0,J0) with vanish
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50 CASIM ABBAS defined as H 1 j (S)
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52 CASIM ABBAS and denoting the cor
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54 CASIM ABBAS formula (2.2) for th
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56 CASIM ABBAS Restricting any of t
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58 CASIM ABBAS where fτ (∞) = li
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60 CASIM ABBAS for τ
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62 CASIM ABBAS Recalling our origin
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64 CASIM ABBAS THEOREM 3.7 Let µ,
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66 CASIM ABBAS Since w2 − w1C 1,
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68 CASIM ABBAS be the conformal tra
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70 CASIM ABBAS so that, for z ∈ B
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72 CASIM ABBAS and, for every R>0,
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74 CASIM ABBAS W 1,p loc (C) since
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Page 83 and 84: 84 GUILLARMOU and TZOU where ∂ν
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Page 95 and 96: 96 GUILLARMOU and TZOU enough, we h
Page 97 and 98: 98 GUILLARMOU and TZOU Using the fa
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Page 111 and 112: 112 GUILLARMOU and TZOU COROLLARY 6
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Page 115 and 116: 116 GUILLARMOU and TZOU Therefore,
Page 117 and 118: 118 GUILLARMOU and TZOU Proof of Pr
Page 119 and 120: 120 GUILLARMOU and TZOU [27] R. B.
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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