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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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