Erdős conjecture on arithmetic progressions

Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

Formally, the conjecture states that if A is a large set in the sense that

then A contains arithmetic progressions of any given length, meaning subsets of the form for arbitrarily large k.

History

In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains infinitely many 3 term arithmetic progressions.[1] This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in what is now known as Szemerédi's theorem.

In a 1976 talk titled "To the memory of my lifelong friend and collaborator Paul Turán," Paul Erdős offered a prize of US$3000 for a proof of this conjecture.[2] As of 2008 the problem is worth US$5000.[3]

Unsolved problem in mathematics:
Does every large set of natural numbers contain arbitrarily long arithmetic progressions?
(more unsolved problems in mathematics)

Erdős' conjecture on arithmetic progressions can be viewed as a stronger version of Szemerédi's theorem. Because the sum of the reciprocals of the primes diverges, the Green–Tao theorem on arithmetic progressions is a special case of the conjecture.

The weaker claim that A must contain infinitely many arithmetic progressions of length 3 is a consequence of an improved bound in Roth's theorem, which appears as the main result in a 2020 preprint by Bloom and Sisask.[4] The former strongest bound in Roth's theorem is due to Bloom.[5]

See also

References
  1. Erdős, Paul; Turán, Paul (1936), "On some sequences of integers" (PDF), Journal of the London Mathematical Society, 11 (4): 261–264, doi:10.1112/jlms/s1-11.4.261.
  2. Problems in number theory and Combinatorics, in Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976), Congress. Numer. XVIII, 35–58, Utilitas Math., Winnipeg, Man., 1977
  3. p. 354, Soifer, Alexander (2008); The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators; New York: Springer. ISBN 978-0-387-74640-1
  4. Bloom, Thomas F.; Sisask, Olof (2020). "Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions". arXiv:2007.03528. Cite journal requires |journal= (help)
  5. Bloom, Thomas F. (2016). "A quantitative improvement for Roth's theorem on arithmetic progressions". Journal of the London Mathematical Society. Second Series. 93 (3): 643–663. arXiv:1405.5800. doi:10.1112/jlms/jdw010. MR 3509957.
  • P. Erdős: Résultats et problèmes en théorie de nombres, Séminaire Delange-Pisot-Poitou (14e année: 1972/1973), Théorie des nombres, Fasc 2., Exp. No. 24, pp. 7,
  • P. Erdős and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
  • P. Erdős: Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Num. Math., Congress Numer. XVIII(1977), 35–58.
  • P. Erdős: On the combinatorial problems which I would most like to see solved, Combinatorica, 1(1981), 28. doi:10.1007/BF02579174
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