## On Roth’s theorem on progressions.(English)Zbl 1264.11004

Klaus Roth [ C. R. Acad. Sci., Paris 234, 388–390 (1952; Zbl 0046.04302)] was the first to prove that a set $$A\subset [1,N]$$ of integers without an arithmetic progression of length $$3$$ satisfies $$|A|=o(N)$$. There is a long and distinguished history of quantitative improvements: D. R. Heath-Brown [J. Lond. Math. Soc., II. Ser. 35, 385–394 (1987; Zbl 0589.10062)]; E. Szemerédi [Acta Math. Hung. 56, No. 1–2, 155–158 (1990; Zbl 0721.11007)]; J. Bourgain [Geom. Funct. Anal. 9, No. 5, 968–984 (1999; Zbl 0959.11004); J. Anal. Math. 104, 155–192 (2008; Zbl 1155.11011)] and Sanders [“On certain other sets of integers”, J. Anal. Math. 116, 53–82 (2012)].
Recently, there has also been great interest in Szemerédi’s generalisation to progressions of length $$k$$. Also, the famous Erdős-Turán conjecture asks whether a set $$A$$ of integers with $$\sum_{a \in A} \frac{1}{a}$$ being divergent must contain an arithmetic progression of length $$k$$.
The author proves that a set $$A\subset [1,N]$$ without 3-progressions satisfies $$|A|= O(\frac{N(\log \log N)^5}{\log N})$$. Quantitatively this appears to be “close” to the Erdős-Turán question, but the author points out that new ideas would be needed to bridge the gap.
The methods involved make use of the Bohr-set technique introduced by Bourgain, and refined by the author, but makes very interesting and novel use of results of N. H. Katz and P. Koester [SIAM J. Discrete Math. 24, No. 4, 1684–1693 (2010; Zbl 1226.05247)], which here is compared to the Dyson $$e$$-transform, and of E. Croot and O. Sisask [Geom. Funct. Anal. 20, No. 6, 1367–1396 (2010; Zbl 1234.11013)]. It can be hoped for that the new ingredients lead to further progress.

### MSC:

 11B25 Arithmetic progressions 11B30 Arithmetic combinatorics; higher degree uniformity
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### References:

  F. A. Behrend, ”On sets of integers which contain no three terms in arithmetical progression,” Proc. Nat. Acad. Sci. U.S.A., vol. 32, pp. 331-332, 1946. · Zbl 0060.10302  J. Bourgain, ”On arithmetic progressions in sums of sets of integers,” in A Tribute to Paul Erd\Hos, Cambridge: Cambridge Univ. Press, 1990, pp. 105-109. · Zbl 0715.11006  J. Bourgain, ”On triples in arithmetic progression,” Geom. Funct. Anal., vol. 9, iss. 5, pp. 968-984, 1999. · Zbl 0959.11004  J. Bourgain, ”Roth’s theorem on progressions revisited,” J. Anal. Math., vol. 104, pp. 155-192, 2008. · Zbl 1155.11011  M. Chang, ”A polynomial bound in Freiman’s theorem,” Duke Math. J., vol. 113, iss. 3, pp. 399-419, 2002. · Zbl 1035.11048  E. Croot and O. Sisask, ”A probabilistic technique for finding almost-periods of convolutions,” Geom. Funct. Anal., vol. 20, iss. 6, pp. 1367-1396, 2010. · Zbl 1234.11013  K. Cwalina and T. Schoen, A linear bound on the dimension in Green-Ruzsa’s theorem, 2010. · Zbl 1304.11122  M. Elkin, ”An improved construction of progression-free sets,” in Symposium on Discrete Algorithms, , 2010, pp. 886-905. · Zbl 1288.11011  B. Green, ”Roth’s theorem in the primes,” Ann. of Math., vol. 161, iss. 3, pp. 1609-1636, 2005. · Zbl 1160.11307  W. T. Gowers and J. Wolf, ”The true complexity of a system of linear equations,” Proc. Lond. Math. Soc., vol. 100, iss. 1, pp. 155-176, 2010. · Zbl 1243.11010  B. Green and J. Wolf, ”A note on Elkin’s improvement of Behrend’s construction,” in Additive Number Theory: Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson, 1st ed., New York: Springer-Verlag, 2010, pp. 141-144. · Zbl 1261.11013  D. R. Heath-Brown, ”Integer sets containing no arithmetic progressions,” J. London Math. Soc., vol. 35, iss. 3, pp. 385-394, 1987. · Zbl 0589.10062  N. H. Katz and P. Koester, ”On additive doubling and energy,” SIAM J. Discrete Math., vol. 24, iss. 4, pp. 1684-1693, 2010. · Zbl 1226.05247  R. Meshulam, ”On subsets of finite abelian groups with no $$3$$-term arithmetic progressions,” J. Combin. Theory Ser. A, vol. 71, iss. 1, pp. 168-172, 1995. · Zbl 0832.11006  K. Roth, ”Sur quelques ensembles d’entiers,” C. R. Acad. Sci. Paris, vol. 234, pp. 388-390, 1952. · Zbl 0046.04302  K. Roth, ”On certain sets of integers,” J. London Math. Soc., vol. 28, pp. 104-109, 1953. · Zbl 0050.04002  T. Sanders, On certain other sets of integers, 2010. · Zbl 1280.11009  T. Schoen, ”Near optimal bounds in Freiman’s theorem,” Duke Math. J., vol. 158, pp. 1-12, 2011. · Zbl 1242.11074  R. Salem and D. C. Spencer, ”On sets of integers which contain no three terms in arithmetical progression,” Proc. Nat. Acad. Sci. U. S. A., vol. 28, pp. 561-563, 1942. · Zbl 0060.10301  T. Schoen and I. Shkredov, Additive properties of multiplicative subgroups in $$\mathbbF_p$$, 2011. · Zbl 1271.11014  E. Szemerédi, ”Integer sets containing no arithmetic progressions,” Acta Math. Hungar., vol. 56, iss. 1-2, pp. 155-158, 1990. · Zbl 0721.11007  T. Tao and V. Vu, Additive Dombinatorics, Cambridge: Cambridge Univ. Press, 2006, vol. 105. · Zbl 1127.11002  R. Yao-Feng and L. Han-Ying, ”On the best constant in Marcinkiewicz-Zygmund inequality,” Stat. Prob. Lett., vol. 53, pp. 227-233, 2001. · Zbl 0991.60011
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