NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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10 TOMASZ SCHOEN THEOREM 10 There is a positive constant c such that, for every finite set A ⊆{1 2 , 2 2 ,...}, we have |A + A| ≥(log |A|) c log log |A| |A|. Proof Set |A + A| =K|A|. By Theorem 8, there is a proper generalized arithmetic pro- C(log K)−1/2 gression P = P1 +···+Pd with d ≤ K and size(P ) ≤ 48K24 |A| such that K)−1/2 −KC(log |A ∩ P |≥e |A|. Suppose that |P1| =maxi |Pi| ≥|P | 1/d , and write P = x∈P2+···+Pd (P1 + x). Now, we make use of a theorem of Bombieri and Zannier [2, Theorem 1] stating that an arithmetic progression of length k contains O(k2/3+ε ) squares. Therefore, we have |A ∩ P |≤ |(P1 + x) ∩ A| ≪ |P1| 3/4 ≤ (48K 24 |A|) 1−1/(4d) , hence x∈P2+···+Pd x∈P2+···+Pd K ≥ (log |A|) c log log |A| , and the assertion follows. ✷ It would be interesting to strengthen Lemma 3 since, clearly, each such improvement would result in better bounds in Freiman’s theorem. The argument presented here has one major weakness—it works only for small sets X (one would prefer to have an analog of this result which is also valid for sets X of roughly the same order as Y ). Indeed, if one could prove a version of Lemma 3 with |X| ∼|Y | and, say ε = 1/20, then Balog-Szemerédi-Gowers’s theorem would give |X ′ +X ′ |≤K 1−c |X ′ |, for some X ′ ⊆ X and c>0. Then, by iterating this process, one would most probably get a result very close to Freiman-Ruzsa’s polynomial conjecture. References [1] Y. BILU, “Structure of sets with small sumset” in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Paris, 1999, 77 – 108. MR 1701189 [2] E. BOMBIERI and U. ZANNIER, A note on squares in arithmetic progressions, II, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13 (2002), 69 – 75. MR 1949479
NEAR OPTIMAL BOUNDS IN FREIMAN’S THEOREM 11 [3] M.-C. CHANG, A polynomial bound in Freiman’s theorem, Duke Math. J. 113 (2002), 399 – 419. MR 1909605 [4] ———, On problems of Erdős and Rudin, J. Funct. Anal. 207 (2004), 444 – 460. MR 2032997 [5] ———, On sum-product representations in Zq, J. Eur. Math. Soc. (JEM) 8 (2006), 435 – 463. MR 2250167 [6] ———, Some consequences of the polynomial Freiman-Ruzsa conjecture, C. R. Math. Acad. Sci. Paris 347 (2009), 583 – 588. MR 2532910 2101, 2, 6, 8911 [7] K. CWALINA and T. SCHOEN, Linear bound on dimension in Green-Ruzsa’s theorem, in preparation. [8] G. A. FREĬMAN, Foundations of a Structural Theory of Set Addition, Trans. Math. Monog. 37, Amer. Math. Soc., Providence, 1973. MR 0360496 [9] W. T. GOWERS, A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529 – 551. MR 1631259 [10] ———, “Rough structure and classification” in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, 79 – 117. MR 1826250 [11] ———, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465 – 588. MR 1844079 [12] B. GREEN, “Finite field models in additive combinatorics” in Surveys in Combinatorics 2005, London Math. Soc. Lecture Note Ser. 327, Cambridge Univ. Press, Cambridge, 2005, 1 – 27. MR 2187732 [13] B. GREEN and I. Z. RUZSA, Freiman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc. (2) 75 (2007), 163 – 175. MR 2302736 [14] B. GREEN and T. TAO, An equivalence between inverse sumset theorems and inverse conjectures for the U 3 norm, Math. Proc. Cambridge. Philos. Soc. 149 (2010), 1 – 19. MR 2651575 [15] N. H. KATZ and P. KOESTER, On additive doubling and energy, preprint, arXiv:0802.4371v1 [math.CO] [16] S. KONYAGIN and I. ŁABA, Distance sets of well-distributed planar sets for polygonal norms, Israel J. Math. 152 (2006), 157 – 175. MR 2214458 [17] T. ŁUCZAK and T. SCHOEN, On a problem of Konyagin, Acta Arith. 134 (2008), 101 – 109. MR 2429639 [18] I. Z. RUZSA, Arithmetical progressions and the number of sums, Period. Math. Hungar. 25 (1992), 105 – 111. MR 1200845 [19] ———, Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379 – 388. MR 1281447 [20] T. SANDERS, Additive structures in sumsets, Math. Proc. Cambridge. Philos. Soc. 144 (2008), 289 – 316. MR 2405891 [21] ———, Appendix to Roth’s theorem on progressions revisited by J. Bourgain, J. Anal. Math. 104 (2008), 193 – 206. MR 2403434 [22] ———, On a nonabelian Balog-Szemerédi-type lemma, J. Aust. Math. Soc. 89 (2010), 127 – 132. MR 2727067 [23] ———, Structure in sets with logarithmic doubling, to appear in Canad. Math. Bull., preprint, arXiv:1002.1552v1 [math.CA]
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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