NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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<strong>IN</strong>HOMOGENEOUS QUADRATIC FORMS 131 Theorem 1.6 is proved using the following, which is a result of combining Theorems 3.1, A.3, andA.4. We have the following. <strong>THEOREM</strong> 3.2 ([EMM1, Theorem 3.5]) Suppose that p ≥ 3 and q ≥ 1. Let f and fˆ be as above. Let ν be any continuous function on K. Then, for every compact subset D of G/ Ɣ, there exist finitely many points x1,...,xℓ ∈ G/ Ɣ such that (i) (ii) the orbit Hxi is closed and has finite H -invariant measure for all i; for any compact set F ⊂ D \ i Hxi, there exists t0 > 0 such that, for all x ∈ F and t>t0, we have f ˆ(atkx) ν(k) dk − fdµ ˆ νdk ≤ ε, (31) K G/ Ɣ K where either µ is the G-invariant measure on G/ Ɣ or H ⋉R n x is closed and has H ⋉R n -invariant probability measure and µ is this measure. Proof We may and will assume that φ is nonnegative. Now define A(r) = ∈ G/ Ɣ : α1() >r . (32) Let gr be a continuous function on G/ Ɣ such that gr() = 0 if /∈ A(r),gr() = 1 for all ∈ A(r + 1), and0≤ gr() ≤ 1 if r ≤ α1() ≤ r + 1. We have f ˆ = ( fˆ − fgr) ˆ + fgr. ˆ Note that fˆ − fgr ˆ is a continuous function with compact support on G/ Ɣ. Note that H 0 ⋉Rn is a maximal connected subgroup of G. Hence for every δ>0, there exists r0 such that if H ⋉Rny is a closed orbit of H ⋉Rn in G/ Ɣ with an H ⋉Rn-invariant probability measure σ ,thenσ (A(r) ∩ H ⋉Rny) r0. Hence for r sufficiently large, we get fdµ− ˆ ( fˆ − fgr) ˆ dµ
132 MARGULIS and MOHAMMADI Hence, if we apply Theorem 3.1, then there is a constant B depending on B1 such that we get ( fgr)(atkx) ˆ ν(k) dk ≤ B sup |ν(k)| r −β/2 (35) K for all x ∈ D. Now choose r>r0sufficiently large so that B(supk∈K |ν(k)|)r−β/2
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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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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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