NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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<strong>IN</strong>HOMOGENEOUS QUADRATIC FORMS 123 Recall from [EMM2]thatifQ is irrational of signature (2, 2), then it has at most four rational null subspaces. Let ÑQ,(a,b,T ) = # x ∈ Z n : Eskin, Margulis, and Mozes proved the following. x is not in a null subspace of Q . (6) x ∈ T and a
124 MARGULIS and MOHAMMADI As in [EMM1], we also have the following uniform version of Theorem 1.5. Let I(p, q) denote the space of inhomogeneous quadratic forms whose homogeneous parts are quadratic forms of signature (p, q) and discriminant ±1. <strong>THEOREM</strong> 1.6 Let D be a compact subset of I(p, q) with p ≥ 3 and q ≥ 1, and let n = p + q. Then for every interval (a,b) and every θ>0, there exists a finite subset P of D such that each Qξ ∈ P is a rational form and, for every compact subset F ⊂ D \ P , there exists T0 such that, for all Qξ ∈ F and T ≥ T0, we have (1 − θ)λQ,(b − a)T n−2 ≤ NQ,ξ,(a,b,T ) ≤ (1 + θ)λQ,(b − a)T n−2 , (10) where λQ, is as in (3). The proofs of the above theorems are straightforward inhomogeneous versions of the arguments and ideas developed in [DM] and[EMM1]. As we mentioned before, Theorem 1.5 fails in signature (2, 2) and (2, 1). In this, paper, we prove an inhomogeneous version of Theorem 1.3. Indeed,asin[EMM2], one needs to assume certain Diophantine conditions on the quadratic form. Using similar ideas, we also give a partial result in the (2, 1) case (see Theorem 1.10 below). We start with the following definitions. Definition 1.7 Let κ>0. A vector ξ = (ξ1,...,ξn) ∈ R n is called κ-Diophantine if there exist C = C(ξ) > 0 such that, for all 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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Page 89 and 90: 90 GUILLARMOU and TZOU LEMMA 2.2.4
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Page 93 and 94: 94 GUILLARMOU and TZOU At a point (
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 103 and 104: 104 GUILLARMOU and TZOU factor, or
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Page 115 and 116: 116 GUILLARMOU and TZOU Therefore,
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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