NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

More documents

Recommendations

Info

4 TOMASZ SCHOEN By α, we denote the distance from α to the nearest integer. For Ɣ ⊆ Zn and γ>0, the Bohr set B(Ɣ, γ ) is defined as B(Ɣ, γ ) ={x ∈ Zn : gx/n ≤γ for every g ∈ Ɣ}. 3. The main lemma The key idea of our approach is to identify a relatively large subset of A that has better additive properties than A. One possible way would be to find A ′ ⊆ A such that |kA ′ |≤K εk |A ′ | for not too-big k. A result of this kind was proved in [17, Lemma 1], but under a stronger assumption on A. Unfortunately, this method does not seem to work in our general setting. Let us remark, however, that Bogolyubov-Ruzsa’s lemma uses the number of additive quadruples rather than the size of the subset. This suggests the following approach: using the assumption |A + A| ≤K|A|, one should find for some small k a relatively dense set X, Y ⊆ kA − kA with E(X, Y ) much bigger than one can expect from the size of A + A. As far as we know, the first such result was proved by Katz and Koester [15]. They showed that there is B ⊆ A ± A, |B| ≫|A|/K C with E(B,B) ≫|B| 3 /K 36/37 . Lemma 3 provides a much better estimate for E(X, Y ) in a “highly nonsymmetric” case, when one set is large and the other is small (a similar result for a diagonal case would give a much better result—see our concluding remarks below). Finally, let us also mention that Sanders in [22], and [23] (seealso[24] for related theorems), used a similar result for a different purpose—that of improving the bounds in many summed versions of Roth-Meshulam’s theorem. LEMMA 3 Suppose that A is a subset of an Abelian group and that |A+A| ≤K|A|. Then for every ε>0, there are sets X ⊆ A and Y ⊆ A + A such that |X| ≥(2K2 ) −21/ε|A|, |Y |≥ |A|, and E(X, Y ) ≥ K−2ε |X| 2 |Y |. Proof For a set B ⊆ A, denote by D = D(B) the set of all t ∈ B − B which has at least |B| 2 |B|2 |B|2 ≥ ≥ 2|B − B| 2|A − A| 2K2 |A| representations as b − b ′ , b,b ′ ∈ B (the last inequality follows from Plünnecke- Ruzsa’s inequality). Note that |D| ≥|B|/2. We first show that there exists a set B ⊆ A such that |B| ≥(2K 2 ) −21/ε +1 |A| and |A + Bt| ≥K −ε |A + B| (2) for all t ∈ D = D(B), where Bt = B ∩ (B + t). We construct B in the following iterative procedure. We start with A 1 = A. Now suppose that A l with
NEAR OPTIMAL BOUNDS IN FREIMAN’S THEOREM 5 |Al |≥(2K2 ) −2l−1 +1 |A| has already been defined. If there exists t0 ∈ D(Al ) with |A + Al t0 |
Page 1 and 2: NEAR OPTIMAL BOUNDS IN FREIMAN’S
Page 3: NEAR OPTIMAL BOUNDS IN FREIMAN’S
Page 13 and 14: SUR LA NON-DENSITÉ DES POINTS ENTI
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
Page 35 and 36: 36 CASIM ABBAS punctured surfaces w
Page 37 and 38: 38 CASIM ABBAS that is, the central
Page 39 and 40: 40 CASIM ABBAS even have A(r) ≡ 0
Page 41 and 42: 42 CASIM ABBAS are contact forms on
Page 43 and 44: 44 CASIM ABBAS the standard complex
Page 45 and 46: 46 CASIM ABBAS and Jδε2 = xγ1(r)
Page 47 and 48: 48 CASIM ABBAS (λ0,J0) with vanish
Page 49 and 50: 50 CASIM ABBAS defined as H 1 j (S)
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,
Page 67 and 68:
68 CASIM ABBAS be the conformal tra
Page 69 and 70:
70 CASIM ABBAS so that, for z ∈ B
Page 71 and 72:
72 CASIM ABBAS and, for every R>0,
Page 73 and 74:
74 CASIM ABBAS W 1,p loc (C) since
Page 75 and 76:
76 CASIM ABBAS (follows from 0 =
Page 77 and 78:
78 CASIM ABBAS is finite, the image
Page 79 and 80:
80 CASIM ABBAS Converting from coor
Page 81 and 82:
82 CASIM ABBAS [27] D. MCDUFF and D
Page 83 and 84:
84 GUILLARMOU and TZOU where ∂ν
Page 85 and 86:
86 GUILLARMOU and TZOU Carleman est
Page 87 and 88:
88 GUILLARMOU and TZOU where F ⊂
Page 89 and 90:
90 GUILLARMOU and TZOU LEMMA 2.2.4
Page 91 and 92:
92 GUILLARMOU and TZOU THEOREM 2.3.
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
Page 99 and 100:
100 GUILLARMOU and TZOU certain exp
Page 101 and 102:
102 GUILLARMOU and TZOU −1 where
Page 103 and 104:
104 GUILLARMOU and TZOU factor, or
Page 105 and 106:
106 GUILLARMOU and TZOU Proof of Pr
Page 107 and 108:
108 GUILLARMOU and TZOU where I1 =
Page 109 and 110:
110 GUILLARMOU and TZOU Proof of Le
Page 111 and 112:
112 GUILLARMOU and TZOU COROLLARY 6
Page 113 and 114:
114 GUILLARMOU and TZOU in x−k−
Page 115 and 116:
116 GUILLARMOU and TZOU Therefore,
Page 117 and 118:
118 GUILLARMOU and TZOU Proof of Pr
Page 119 and 120:
120 GUILLARMOU and TZOU [27] R. B.
Page 121 and 122:
122 MARGULIS and MOHAMMADI multiple
Page 123 and 124:
124 MARGULIS and MOHAMMADI As in [E
Page 125 and 126:
126 MARGULIS and MOHAMMADI planes
Page 127 and 128:
128 MARGULIS and MOHAMMADI α ∈ R
Page 129 and 130:
130 MARGULIS and MOHAMMADI We fix s
Page 131 and 132:
132 MARGULIS and MOHAMMADI Hence, i
Page 133 and 134:
134 MARGULIS and MOHAMMADI Hence th
Page 135 and 136:
136 MARGULIS and MOHAMMADI Indeed,
Page 137 and 138:
138 MARGULIS and MOHAMMADI For simp
Page 139 and 140:
140 MARGULIS and MOHAMMADI Note tha
Page 141 and 142:
142 MARGULIS and MOHAMMADI (ii) The
Page 143 and 144:
144 MARGULIS and MOHAMMADI needs to
Page 145 and 146:
146 MARGULIS and MOHAMMADI Proof Le
Page 147 and 148:
148 MARGULIS and MOHAMMADI all j>s,
Page 149 and 150:
150 MARGULIS and MOHAMMADI 0 c
Page 151 and 152:
152 MARGULIS and MOHAMMADI Some imp
Page 153 and 154:
154 MARGULIS and MOHAMMADI of G, an
Page 155 and 156:
156 MARGULIS and MOHAMMADI (ii) for
Page 157 and 158:
158 MARGULIS and MOHAMMADI PROPOSIT
Page 159:
160 MARGULIS and MOHAMMADI [EMM2]
show all

NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

Create successful ePaper yourself

Delete template?

Save as template?