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.


11B25 Arithmetic progressions
11B30 Arithmetic combinatorics; higher degree uniformity
Full Text: DOI arXiv


[1] 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
[2] 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
[3] J. Bourgain, ”On triples in arithmetic progression,” Geom. Funct. Anal., vol. 9, iss. 5, pp. 968-984, 1999. · Zbl 0959.11004
[4] J. Bourgain, ”Roth’s theorem on progressions revisited,” J. Anal. Math., vol. 104, pp. 155-192, 2008. · Zbl 1155.11011
[5] M. Chang, ”A polynomial bound in Freiman’s theorem,” Duke Math. J., vol. 113, iss. 3, pp. 399-419, 2002. · Zbl 1035.11048
[6] 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
[7] K. Cwalina and T. Schoen, A linear bound on the dimension in Green-Ruzsa’s theorem, 2010. · Zbl 1304.11122
[8] M. Elkin, ”An improved construction of progression-free sets,” in Symposium on Discrete Algorithms, , 2010, pp. 886-905. · Zbl 1288.11011
[9] B. Green, ”Roth’s theorem in the primes,” Ann. of Math., vol. 161, iss. 3, pp. 1609-1636, 2005. · Zbl 1160.11307
[10] 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
[11] 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
[12] 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
[13] 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
[14] 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
[15] K. Roth, ”Sur quelques ensembles d’entiers,” C. R. Acad. Sci. Paris, vol. 234, pp. 388-390, 1952. · Zbl 0046.04302
[16] K. Roth, ”On certain sets of integers,” J. London Math. Soc., vol. 28, pp. 104-109, 1953. · Zbl 0050.04002
[17] T. Sanders, On certain other sets of integers, 2010. · Zbl 1280.11009
[18] T. Schoen, ”Near optimal bounds in Freiman’s theorem,” Duke Math. J., vol. 158, pp. 1-12, 2011. · Zbl 1242.11074
[19] 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
[20] T. Schoen and I. Shkredov, Additive properties of multiplicative subgroups in \(\mathbbF_p\), 2011. · Zbl 1271.11014
[21] E. Szemerédi, ”Integer sets containing no arithmetic progressions,” Acta Math. Hungar., vol. 56, iss. 1-2, pp. 155-158, 1990. · Zbl 0721.11007
[22] T. Tao and V. Vu, Additive Dombinatorics, Cambridge: Cambridge Univ. Press, 2006, vol. 105. · Zbl 1127.11002
[23] 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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