NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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<strong>IN</strong>HOMOGENEOUS QUADRATIC FORMS 149 This finishes the proof of Theorem 4.1. 7. Proof of Theorem 1.10 The proof of Theorem 1.10 is very similar to that of Theorem 1.9. Indeed, our study in this case is simpler as we are dealing with the case where the homogeneous part Q is rational. Hence, we only need to consider the contribution of null subspaces to the counting function. As before, let q ∈ SL3(R) be such that Q(v) = B(qv) for all v ∈ R 3 . Since Q is rational, we may assume that q is in PGL3(Q). Let p ∈ GL3(Q) be a representative for q. Let = qZ 3 , and define qξ = q(Z 3 + ξ). As in Section 4, letX(qξ) be the set of vectors in qξ not contained in qL, where L ⊂ Z 3 is an exceptional (1-dimensional) subspace for Q. An argument like that in Lemma 6.1 shows that there are at most three such subspaces when Q is rational and that ξ is an irrational vector. For any continuous compactly supported function f on R 3 , define f ˜(g : qξ) = v∈X(qξ ) f (gv). (74) As in previous discussions, we reduce the proof of Theorem 1.10 to the following theorem. <strong>THEOREM</strong> 7.1 Let G, H, K, and {at} be as in Section 2 for the signature (2, 1) case. Let Qξ be a quadratic form of signature (2, 1) as in the statement of Theorem 1.10.Letq∈ SL3(R) and qξ be as above. Let ν be a continuous function on K. Then we have lim sup t→∞ K ˜f (atk : qξ)ν(k) dk ≤ G/ Ɣ f ˆ(g) dµ(g) K ν(k) dk, (75) where µ is the H ⋉R 3 -invariant probability measure on the closed orbit H ⋉R 3 ·qξ. The proof of this theorem is very similar to that of Theorem 4.1. We will use the same notations as those in the previous sections for the sake of simplicity. The main notational difference to bear in mind is that in previous sections L denotes a 2dimensional subspace, where in this section L is a 1-dimensional (null) subspace. As before, we need to study the subsets of K where the function ˜f is “large”. We start by recalling the following. There is c>0 such that, for all large t and small
150 MARGULIS and MOHAMMADI 0 c ⊂ k ∈ K : α1(atkqξ) δ > 1 ∪ k ∈ K : α2(atkqξ) > δ 1 . (76) δ We fix t and δ as above. Let us make two important remarks before we continue. Remarks 7.2 (i) Using reduction theory of the orthogonal group, we see that if α2(atkqξ) > 1/δ, then actually α1(atkqξ) > 1/δ. Hence we only need to study the contribution coming from α1. (ii) The fact that Q is a rational form implies that there exists δ0 depending only on Q such that if 0
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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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76 CASIM ABBAS (follows from 0 =
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78 CASIM ABBAS is finite, the image
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80 CASIM ABBAS Converting from coor
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82 CASIM ABBAS [27] D. MCDUFF and D
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84 GUILLARMOU and TZOU where ∂ν
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86 GUILLARMOU and TZOU Carleman est
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88 GUILLARMOU and TZOU where F ⊂
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90 GUILLARMOU and TZOU LEMMA 2.2.4
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92 GUILLARMOU and TZOU THEOREM 2.3.
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94 GUILLARMOU and TZOU At a point (
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96 GUILLARMOU and TZOU enough, we h
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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