Duke Mathematical Journal

A polynomial bound in Freiman's theorem

Mei-Chu Chang

Abstract

In this paper the following improvement on Freiman's theorem on set addition is obtained (see Theorems 1 and 2 in Section 1).

Let $A\subset \mathbb {Z}$ be a finite set such that $|A+A|<\alpha|A|$. Then A is contained in a proper d-dimensional progression P, where $d\leq [\alpha -1]$ and $\log (|P|/|A|).

Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa's method, which we improve in several places, as well as Y. Bilu's proof of Freiman's theorem.

Article information

Source
Duke Math. J., Volume 113, Number 3 (2002), 399-419.

Dates
First available in Project Euclid: 18 June 2004

https://projecteuclid.org/euclid.dmj/1087575313

Digital Object Identifier
doi:10.1215/S0012-7094-02-11331-3

Mathematical Reviews number (MathSciNet)
MR1909605

Zentralblatt MATH identifier
1035.11048

Citation

Chang, Mei-Chu. A polynomial bound in Freiman's theorem. Duke Math. J. 113 (2002), no. 3, 399--419. doi:10.1215/S0012-7094-02-11331-3. https://projecteuclid.org/euclid.dmj/1087575313

References

• V. Bergelson and A. Liebman, Polynomial extensions of van der Waerden's and Szemerédi's theorem, J. Amer. Math. Soc. 9 (1996), 725--753.
• Y. Bilu, Structure of sets with small sumset'' in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Montrouge, 1999, 77--108.
• N. N. Bogolioùboff [Bogolyubov], Sur quelques propriétés arithmétiques des presque-periodes, Ann. Chaire Phys. Math. Kiev 4 (1939), 185--205.
• M. Chaimovich, New structural approach to integer programming: A survey'' in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Montrouge, 1999, 341--362.
• M.-C. Chang, Inequidimensionality of Hilbert schemes, Proc. Amer. Math. Soc. 125 (1997), 2521--2526.
• G. Cohen and G. Zémor, Subset sums and coding theory'' in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Montrouge, 1999, 327--339.
• G. Elekes, On the number of sums and products, Acta Arith. 81 (1997), 365--367.
• P. Erdős and E. Szemerédi, On sums and products of integers'' in Studies in Pure Mathematics, Birkhäuser, Basel, 1983, 213--218.
• K. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math. 85, Cambridge Univ. Press, Cambridge, 1986.
• G. Freiman, Foundations of a Structural Theory of Set Addition, Trans. Math. Monogr. 37, Amer. Math. Soc., Providence, 1973.
• G. Freiman, H. Halberstam, and I. Ruzsa, Integer sum sets containing long arithmetic progressions, J. London Math. Soc. (2) 46 (1992), 193--201.
• H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204--256.
• H. Furstenberg, Y. Katznelson, and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 527--552.
• W. T. Gowers, A new proof of Szemerédi's theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529--551.
• --. --. --. --., A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001), 465--588. \CMP1 844 079
• M. Herzog, New results on subset multiplication in groups'' in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Montrouge, 1999, 309--315.
• N. Katz, I. Laba, and T. Tao, An improved bound on the Minkowski dimension of Besicovitch sets in $R^3$, Ann. of Math. (2) 152 (2000), 383--446. \CMP1 804 528
• N. Katz and T. Tao, Some connections between Falconer's distance set conjecture and sets of Furstenburg type, New York J. Math. 7 (2001), 149--187. \CMP1 856 956
• J. Lopez and K. Ross, Sidon Sets, Lecture Notes in Pure and Appl. Math. 13, Dekker, New York, 1975.
• M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer, New York, 1996.
• M. Nathanson and G. Tenenbaum, Inverse theorems and the number of sums and products'' in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Montrouge, 1999, 195--204.
• K. Roth, On certain sets of integers, J. London Math. Soc 28 (1953), 104--109.
• W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203--227.
• I. Ruzsa, Arithmetic progressions in sumsets, Acta Arith. 60 (1991), 191--202.
• --. --. --. --., Generalized arithmetic progressions and sumsets, Acta Math. Hungar. 65 (1994), 379--388.
• --. --. --. --., An analog of Freiman's theorem in groups'' in Structure Theory of Set Addition, Astérisque 258, Soc. Math. France, Montrouge, 1999, 323--326.
• S. Shelah, Primitive recursive bounds for van der Waerden numbers, J. Amer. Math. Soc. 1 (1988), 683--697.
• E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199--245.
• E. Szemerédi and W. Trotter, Extremal problems in discrete geometry, Combinatorica 3 (1983), 381--392.
• T. Tao, From rotating needles to stability of waves: Emerging connections between combinatorics, analysis, and PDE, Notices Amer. Math. Soc. 48 (2001), 294--303.
• B. L. van der Waerden, Beweis einer Baudetsche Vermutung, Nieuw Arch. Wisk. 15 (1927), 212--216.
• T. Wolff, Recent work connected with the Kakeya problem'' in Prospects in Mathematics (Princeton, 1996), Amer. Math. Soc., Providence, 1999, 129--162.