NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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<strong>IN</strong>HOMOGENEOUS QUADRATIC FORMS 157 for any x ∈ F ⊂ D \ i H ⋉Rnxi. Now if we apply [EMM1, Lemma 4.2], then there exists an open subset W ⊂ G/ Ɣ such that i CiƔ/ Ɣ ⊂ W and such that |{k ∈ K : kx ∈ W }| 0 such that 1 T φ(uty) − φdg 0 be any sufficiently small number, and let T > 2. There exists a rational 3-dimensional subspace U of 2 R 4 with a reduced integral basis of norm at most T τ whose projections into V2 have norm less than T τ−1 such that one of the following holds. (a) The restriction of Q (6) to U is anisotropic over Q, in which case (i) in Theorem 5.6 holds. (b) The restriction of Q (6) to U splits over Q, in which case (ii) in Theorem 5.6 holds. Furthermore, in this case Q ′ (as in loc. cit.) is proportional to f (U,U ⊥ ), where U ⊥ is the orthogonal complement of U with respect to Q (6) and where f is as in Lemma 4.3. Moreover, the number of quasi-null subspaces with norm between T/2 and T which are not in U or U ⊥ is O(T 1−τ ). Let us denote by Q (3) (v) the restriction of Q (6) to U. As we mentioned before, this is a form with signature (2, 1) and U is a rational subspace. If L is a quasi-null subspace with T/2 ≤v L ≤T, then we have Q (6) (v L ) = 0. Hence if L is a quasi-null subspace in U, then Q (3) (v L ) = 0. In other words, Proposition 6.2 follows from the following.
158 MARGULIS and MOHAMMADI PROPOSITION B.2 Let Q0(x,y,z) = 2xz − y 2 be the standard quadratic form of signature (2, 1) on R 3 . Let = gZ 3 ,whereg ∈ GL3(Q). Let T > 0 be a large parameter, and let τ be a sufficiently small parameter. Assume that the entries of g are rational numbers whose nominator and denominators are bounded by a fixed power of T τ , and also suppose that |det g| ≤T τ . Further assume that the length of the shortest vector in is c = O(1). Then if T is large enough, we have # w ∈ P () : Q0(w) = 0 and w ≤T 0. Note that C is dilation-invariant. Let λ = disc() 1/3 , and define 1 = 1/λ. The lattice 1 is unimodular, and the length of the shortest vector in 1 is at least c/λ. Let¯g ∈ PGL3(Q) denote the image of g. Indeed, 1 = ¯gZ 3 . The counting problem in (98) follows if we show that # w ∈ P (1) : Q0(w) = 0 and w ≤ T
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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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98 GUILLARMOU and TZOU Using the fa
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100 GUILLARMOU and TZOU certain exp
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102 GUILLARMOU and TZOU −1 where
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104 GUILLARMOU and TZOU factor, or
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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