NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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<strong>IN</strong>HOMOGENEOUS QUADRATIC FORMS 129 Quadratic forms. Let n ≥ 3, andletn = p + q, where p ≥ 2. Let {e1,...,en} be the standard basis for R n . If p ≥ 3, let B be the “standard" form i B i=1 xiei = 2x1xn + p i=2 x 2 i − n−1 i=p+1 x 2 i . (23) Let H = SO(B),andlet{at} be the 1-parameter subgroup of H given by ate1 = e −t e1, atei = ei for 2 ≤ i ≤ n − 1, andaten = e t en. And let K = H ∩ ˆK, where ˆK is the group of orthogonal matrices with determinant 1. We let dk denote the Haar measure on K normalized so that K is a probability space. If (p, q) = (2, 2), welet B(x1,x2,x3,x4) = x1x4 − x2x3 be the standard form on R4 . This is the determinant on M2(R) if we identify R4 with M2(R). Note that this identification shows that SO(2, 2) is locally isomorphic to SL2(R) × SL2(R) with the action v → g1vg −1 2 , which leaves the determinant invariant. We let H = SL2(R) × SL2(R), K = SO(2) × SO(2), andat = (bt,bt), where bt = diag(e −t/2 ,e t/2 ). We let dk denote the Haar measure on K normalized so that K is a probability space. We often work with the standard lattice Z 4 and the form Q, in which case we continue to denote by {at} and K the corresponding 1-parameter and maximal compact subgroup of SO(Q). If (p, q) = (2, 1), welet B(x1,x2,x3) = x1x3 − x 2 2 be the standard form on R3 . This is the determinant on Sym2(R), the space of 2 × 2 symmetric matrices, if we identify R3 with Sym2(R). This identification shows that SO(2, 1) is locally isomorphic to SL2(R) with the action v → gvtg, where tg is the transpose matrix. We let H = SL2(R). We let at = diag(e−t/2 ,et/2 ),andwe let K = SO(2) be the maximal compact subgroup of H. As before, dk denotes the normalized Haar measure on K. Let f be a continuous function with compact support on Rn . We define the theta transform of f by f ˆ( + ξ) = f (v), (26) v∈+ξ where +ξ is any unimodular inhomogeneous lattice in R n . Note that ˆ f is a function on the space of inhomogeneous lattices. (24) (25)
130 MARGULIS and MOHAMMADI We fix some more notations. Let n = p + q, andletG = SLn(R) ⋉R n . Let Ɣ = SLn(Z) ⋉Z n , which is a lattice in G. We have the following, which is similar to Siegel’s integral formula. LEMMA 2.1 Let f and fˆ be as above. Let µ be a probability measure on G/ Ɣ which is invariant under Rn . Then f ˆ(g) dµ(g) = G/ Ɣ Rn f (x) dx. (27) Proof Note that Rn is the unipotent radical of G. Now the lemma follows from Fubini’s theorem and the fact that µ is Rn-invariant. We end this section by recalling the definition of the α functions defined on the space of lattices. Let be a lattice in R n . A subspace L of R n is called -rational if L ∩ is a lattice in L. For a -rational subspace L, letd(L) be the volume of L/(L ∩ ). For 0 ≤ i ≤ n,define 1 αi() = sup d(L) : L is a -rational subspace of dimension i , (28) and let α() = maxi αi(). Now if ξ = + ξ is an inhomogeneous lattice, let α(ξ) = α(). There is a constant c = c(f ) depending on f such that, for any inhomogeneous lattice ξ = + ξ, wehave ˆ f (ξ)
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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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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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