NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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<strong>IN</strong>HOMOGENEOUS QUADRATIC FORMS 141 let aφ =(at(kθ,kφ)[λ0 + ζ ]) Lφ ⊥. Recall that T/2 ≤vL ≤T, hence aT/2 ≤ w ≤aT. As was mentioned above, K2 acts on the plane spanned by {e123,e124} by rotation on R 2 , and e123 is the contracting direction of {at}. Also note that we are assuming that |E L t (λ0)| >δ η2 . Hence, there exist kφ ∈ E L t (λ0) such that aT δ η2 2 ≤ at(kθ,kφ)[v L ∧ (λ0 + ζ )] = aφat(kθ,kφ)v L . (57) Note now that we have aφ ≤ r and at(kθ,kφ)v L ≤δ. So we get a ≤ (2rδ 1−η2 )/T, as we wanted to show. Before proceeding, let us draw the following corollary. This will be used in the proof of Theorem 4.1 to control the contribution of “small” subspaces. COROLLARY 5.5 Let η1 be as in Proposition 5.3, and let η2 1, there is a δ0 = δ0(M,) such that if δ2, one of the following holds. (i) The number of quasi-null subspaces of Q of norm between T/2 and T is O(T 1−τ2 ).
142 MARGULIS and MOHAMMADI (ii) There exists a split integral form Q ′ with coefficients bounded by a fixed power of T τ2 and 1 ≤ λ ∈ R satisfying Q − (1/λ)Q ′ ≤T −ρ such that the number of quasi-null subspaces of Q with norm between T/2 and T which are not null subspaces of Q ′ is O(T 1−τ2 ). We refer to [EMM2, Section 10] and also Section 7 of this paper for a more careful analysis of this theorem. The main ingredient in the proof is the system of inequalities from [EMM1]. The main difference is that these inequalities are applied to a certain dilated lattice in 2 R 4 = R 6 . Let us now outline the rest of the proof. Let Qξ be the inhomogeneous quadratic form as in the statement of Theorem 1.9. We apply the above theorem with appropriate parameters ρ,τ2 to be determined later. Let T ≥ 2 be given. Now if (i) above holds (which is always the case if Q is not EWAS), then we already have a good control on the number of quasi-null subspaces in question, and we will get the desired control on the measure of the set L AL t (δ, ε). Hence, we may assume that (ii) above holds. Again, if L is not a null subspace of the appropriate approximation of Q, we proceed as in case (i). So we need to consider the contribution from quasi-null subspaces which are null subspaces of some rational approximation. In this case, using Lemma 5.3,we are reduced to the case where only one translate of L has contribution. We then use the Diophantine property of ξ, and we get a control on the number of such subspaces. This completes the proof. We need to fix some more notations before proceeding with the above outline. If Q is a rational form, we may choose µ1 small enough such that all µ1-quasi-null subspaces are null subspaces. Also in this case, by replacing Q with a scalar multiple, we may and will assume that Q is a primitive integral form. Let T ≥ 2 be a fixed number. Recall from Theorem 5.6 that there are two possibilities for quasi-null subspaces. The case which requires more study is case (ii), so let us assume that we are in this case. Let QT = Q ′ (resp., QT = Q) ifQ is an irrational form (resp., if Q is split integral form), where Q ′ is given as in (ii) of Theorem 5.6. LetQt(δ, ε, τ2) be the set of all such quasi-null subspaces (resp., null subspaces if Q is an rational) which are not exceptional subspaces and such that A2 δ (•,ε) is nonempty for them. Let L ∈ Qt(δ, ε, τ2) be such a subspace. We further assume that L is of the first type and that T/2 ≤vL≤T. Since QT is a split integral form, after possibly multiplying by a scalar bounded by a fixed power of T τ2 , there exists a nonsingular integral matrix such that QT (v) = B(pv). Furthermore, Theorem 5.6 guarantees that we may choose p such that its entries are bounded by a fixed power of T τ2 . Recall that the null subspaces of B are of two types based on the fact that the corresponding vector vL is in V1 or V2. From now on by a null space of first kind for B, we mean the following.
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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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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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