NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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26 PASCAL AUTISSIER I(N) ⊂ I1(N):sib ∈ N et c ∈ Nr − N, alors il existe un indice i tel que bi ≥ ci + 1, donc ϕ b1 1 ,...,ϕbr r ∈ ϕci+1 i A ⊂〈ϕc1+1 1 ,...,ϕcr +1 r 〉. On prouve l’inclusion I1(N) ⊂ I(N) par récurrence sur r. Le cas r = 1 est facile (avec le lemme 6.1 si N est vide). Supposons r ≥ 2 et le résultat au cran r − 1. Montrons par une (deuxième) récurrence sur k que I1(N) ⊂ I(N) + ϕk 1A pour tout k ∈ N. Sik = 0, cette inclusion est évidente. Supposons k ≥ 1 et le résultat au cran k − 1. Soitsun élément de I1(N). Ils’écrit s = x + ϕ k−1 1 a avec un x ∈ I(N) et un a ∈ A. Posons Nk = {b = (b2,...,br) ∈ Nr−1 | (k − 1,b2,...,br) ∈ N}. Soit b ∈ N r−1 − Nk. Alorsϕ k−1 1 a = s − x appartient à I1(N) donc en particulier à 〈ϕk 1 ,ϕb2+1 2 ,...,ϕbr +1 r 〉. Autrement dit, il existe a ′ ∈ A tel que ϕ k−1 1 a − ϕk 1a′ ∈ 〈ϕ b2+1 2 ,...,ϕbr +1 r 〉. Puisque (ϕ b2+1 2 ,...,ϕbr +1 r ,ϕ1) est une suite régulière de A, on obtient a − ϕ1a ′ ∈〈ϕ b2+1 2 ,...,ϕbr +1 r 〉 en simplifiant par ϕ k−1 1 .Réduisons modulo ϕ1 on trouve dans A ′ = A/ϕ1A que ā ∈〈¯ ϕ2 b2+1 ,..., ϕr ¯ br +1 〉. On a ainsi que ā ∈ b∈Nr−1 〈 ¯ −Nk ϕ2 b2+1 ,..., ϕr ¯ br +1 〉.Or(¯ ϕ2,..., ϕr) ¯ est une suite régulière de A ′ , donc ā ∈〈¯ ϕ2 b2 ··· ϕr ¯ br , b ∈ Nk〉 par la première hypothèse de récurrence. On écrit a = y + ϕ1a1 avec y ∈〈ϕ b2 2 ···ϕbr r ,b∈ Nk〉 et a1 ∈ A. Ilen découle l’égalité s = x + ϕ k−1 1 y + ϕk 1a1 avec ϕ k−1 1 y ∈〈ϕk−1 1 ϕb2 2 ···ϕbr r ,b∈ Nk〉 ⊂ I(N), qui implique bien que s est dans I(N) + ϕk 1A. D’où l’inclusion I1(N) ⊂ k∈N (I(N) + ϕk 1A). On conclut par le lemme 6.1. Remerciements. Je remercie Yuri Bilu pour m’avoir fourni la référence [2]. Je remercie également les rapporteurs pour leurs suggestions pertinentes. References [1] P. AUTISSIER, Géométrie, points entiers et courbes entières, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 221 – 239. MR 2518077 14, 17, 21, 23, 24 [2] P. CORVAJA, A. LEVIN et U. ZANNIER, Integral points on threefolds and other varieties, Tohoku Math. J. 61 (2009), 589 – 601. MR 2598251 14, 26 [3] P. CORVAJA et U. ZANNIER, A subspace theorem approach to integral points on curves, C. R. Math. Acad. Sci. Paris 334 (2002), 267 – 271. MR 1891001 14 [4] ———, On integral points on surfaces, Ann. of. Math. (2) 160 (2004), 705 – 726. MR 2123936 14, 17 [5] W. FULTON, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. 3, Springer, Berlin, 1998. MR 1644323 16 [6] S. LANG, Number Theory, III: Diophantine Geometry, Encyclopaedia Math. Sci. 60, Springer, Berlin, 1991. MR 1112552 13
SUR LA NON-DENSITÉ DES POINTS ENTIERS 27 [7] R. LAZARSFELD, Positivity in Algebraic Geometry, I: Classical Setting—Line Bundles and Linear Series, Ergeb. Math. Grenzgeb. 48, Springer, Berlin, 2004. MR 2095471 23 [8] A. LEVIN, Generalizations of Siegel’s and Picard’s theorems, Ann. of Math. (2) 170 (2009), 609 – 655. MR 2552103 14, 17 [9] H. MATSUMURA, Commutative Algebra, 2nd ed., Math. Lecture Note Ser. 56, Benjamin/Cummings, Reading, Mass., 1980. MR 0575344 25 [10] H. P. SCHLICKEWEI, The p-adic Thue-Siegel-Roth-Schmidt theorem, Arch. Math. (Basel) 29 (1977), 267 – 270. MR 0491529 14 [11] W. M. SCHMIDT, Diophantine approximation, Lecture Notes in Math. 785, Springer Berlin, 1980. MR 0568710 14 [12] J.-P. SERRE, Lectures on the Mordell-Weil Theorem, 3rd ed., Aspects Math., Friedr. Vieweg and Sons, Braunschweig, 1997. MR 1757192 [13] P. VOJTA, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239, Springer, Berlin, 1987. MR 0883451 13, 15 [14] ———, A refinement of Schmidt’s subspace theorem, Amer. J. Math. 111 (1989), 489 – 518. MR 1002010 20 [15] ———, Integral points on subvarieties of semiabelian varieties, I, Invent. Math. 126 (1996), 133 – 181. MR 1408559 14 [16] ———, On Cartan’s theorem and Cartan’s conjecture, Amer. J. Math. 119 (1997), 1 – 17. MR 1428056 23 [17] S. ZHANG, Small points and adelic metrics,J.AlgebraicGeom.4 (1995), 281 – 300. MR 1311351 19 Institut de Mathématiques de Bordeaux, Université Bordeaux I, 33405 Talence CEDEX, France; pascal.autissier@math.u-bordeaux1.fr
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Page 29 and 30: 30 CASIM ABBAS Recall that the Reeb
Page 31 and 32: 32 CASIM ABBAS that Here the energy
Page 33 and 34: 34 CASIM ABBAS • uτ ( ˙S) ∩ u
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Page 45 and 46: 46 CASIM ABBAS and Jδε2 = xγ1(r)
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Page 51 and 52: 52 CASIM ABBAS and denoting the cor
Page 53 and 54: 54 CASIM ABBAS formula (2.2) for th
Page 55 and 56: 56 CASIM ABBAS Restricting any of t
Page 57 and 58: 58 CASIM ABBAS where fτ (∞) = li
Page 59 and 60: 60 CASIM ABBAS for τ
Page 61 and 62: 62 CASIM ABBAS Recalling our origin
Page 63 and 64: 64 CASIM ABBAS THEOREM 3.7 Let µ,
Page 65 and 66: 66 CASIM ABBAS Since w2 − w1C 1,
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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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106 GUILLARMOU and TZOU Proof of Pr
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108 GUILLARMOU and TZOU where I1 =
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110 GUILLARMOU and TZOU Proof of Le
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112 GUILLARMOU and TZOU COROLLARY 6
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114 GUILLARMOU and TZOU in x−k−
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116 GUILLARMOU and TZOU Therefore,
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118 GUILLARMOU and TZOU Proof of Pr
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120 GUILLARMOU and TZOU [27] R. B.
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122 MARGULIS and MOHAMMADI multiple
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124 MARGULIS and MOHAMMADI As in [E
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126 MARGULIS and MOHAMMADI planes
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128 MARGULIS and MOHAMMADI α ∈ R
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130 MARGULIS and MOHAMMADI We fix s
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132 MARGULIS and MOHAMMADI Hence, i
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134 MARGULIS and MOHAMMADI Hence th
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136 MARGULIS and MOHAMMADI Indeed,
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138 MARGULIS and MOHAMMADI For simp
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140 MARGULIS and MOHAMMADI Note tha
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142 MARGULIS and MOHAMMADI (ii) The
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144 MARGULIS and MOHAMMADI needs to
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148 MARGULIS and MOHAMMADI all j>s,
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150 MARGULIS and MOHAMMADI 0 c
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152 MARGULIS and MOHAMMADI Some imp
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154 MARGULIS and MOHAMMADI of G, an
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156 MARGULIS and MOHAMMADI (ii) for
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158 MARGULIS and MOHAMMADI PROPOSIT
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160 MARGULIS and MOHAMMADI [EMM2]
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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