The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view

Kra, Bryna

doi:10.1090/S0273-0979-05-01086-4

The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view

Author: Bryna Kra
Journal: Bull. Amer. Math. Soc. 43 (2006), 3-23
MSC (2000): Primary 11N13, 37A45, 11B25
DOI: https://doi.org/10.1090/S0273-0979-05-01086-4
Published electronically: October 6, 2005
MathSciNet review: 2188173
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Abstract: A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only prog- ress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer $k$, there exist infinitely many arithmetic progressions of length $k$ consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.

Antal Balog, The prime $k$-tuplets conjecture on average, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 47–75. MR 1084173, DOI https://doi.org/10.1007/978-1-4612-3464-7_5
Antal Balog, Linear equations in primes, Mathematika 39 (1992), no. 2, 367–378. MR 1203292, DOI https://doi.org/10.1112/S0025579300015096
V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 337–349. MR 912373, DOI https://doi.org/10.1017/S0143385700004090
V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Amer. Math. Soc. 9 (1996), no. 3, 725–753. MR 1325795, DOI https://doi.org/10.1090/S0894-0347-96-00194-4
Jean-Pierre Conze and Emmanuel Lesigne, Sur un théorème ergodique pour des mesures diagonales, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 12, 491–493 (French, with English summary). MR 939438

ET P. Erdös and P. Turán. On some sequences of integers. J. London Math. Soc., 11 (1936), 261-264.

FJU M. Frind, P. Jobling and P. Underwood. $23$ primes in arithmetic progression. Available at http://primes.plentyoffish.com/

Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 498471, DOI https://doi.org/10.1007/BF02813304
H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291 (1979). MR 531279, DOI https://doi.org/10.1007/BF02790016
H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem, J. Anal. Math. 57 (1991), 64–119. MR 1191743, DOI https://doi.org/10.1007/BF03041066
Hillel Furstenberg and Benjamin Weiss, A mean ergodic theorem for $(1/N)\sum ^N_{n=1}f(T^nx)g(T^{n^2}x)$, Convergence in ergodic theory and probability (Columbus, OH, 1993) Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 193–227. MR 1412607

GY D. Goldston and C. Y. Yildirim. Small gaps between primes, I. Preprint.

W. T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), no. 3, 465–588. MR 1844079, DOI https://doi.org/10.1007/s00039-001-0332-9

Granville A. Granville. Personal communication.

Gr B. Green. Roth’s Theorem in the primes. To appear in Ann. Math.

GT B. Green and T. Tao. The primes contain arbitrarily long arithmetic progressions. To appear, Ann. Math.

GT2 B. Green and T. Tao. A bound for progressions of length $k$ in the primes. Preprint.

HL G. H. Hardy and J. E. Littlewood. Some problems of “partitio numerorum” III: on the expression of a number as a sum of primes. Acta Math., 44 (1923), 1–70.

D. R. Heath-Brown, Three primes and an almost-prime in arithmetic progression, J. London Math. Soc. (2) 23 (1981), no. 3, 396–414. MR 616545, DOI https://doi.org/10.1112/jlms/s2-23.3.396
Bernard Host and Bryna Kra, Convergence of Conze-Lesigne averages, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 493–509. MR 1827115, DOI https://doi.org/10.1017/S0143385701001249

HK B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. Ann. Math., 161 (2005), 397–488.

HK2 B. Host and B. Kra. Convergence of polynomial ergodic averages. Israel J. Math., 149 (2005), 1–19.

L A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math., 146 (2005), 303–316.

M F. Mertens. Ein Beitrag zur analytischen Zahlentheorie. Journal für Math., 78 (1874), 46–62.

Paul A. Pritchard, Andrew Moran, and Anthony Thyssen, Twenty-two primes in arithmetic progression, Math. Comp. 64 (1995), no. 211, 1337–1339. MR 1297475, DOI https://doi.org/10.1090/S0025-5718-1995-1297475-1
Atle Selberg, The general sieve-method and its place in prime number theory, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 1, Amer. Math. Soc., Providence, R. I., 1952, pp. 286–292. MR 0044563
E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245. MR 369312, DOI https://doi.org/10.4064/aa-27-1-199-245

T T. Tao. A quantitative ergodic theory proof of Szemerédi’s theorem. Preprint.

T2 T. Tao. A remark on Goldston-Yildirim correlation estimates. Preprint.

VDC J. G. van der Corput. Über Summen von Primzahlen und Primzahlquadraten. Math. Ann., 116 (1939), 1–50.

VDW B. L. van der Waerden. Beweis einer Baudetschen Vermutung. Nieuw Arch. Wisk., 15 (1927), 212–216.

Z T. Ziegler. Universal characteristic factors and Furstenberg averages. Preprint.