Erdős conjecture on arithmetic progressions explained

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

$$\backslash sum\_\; \backslash frac\; \backslash \; =\backslash \; \backslash infty,$$

then A contains arithmetic progressions of any given length, meaning that for every positive integer k there are an integer a and a non-zero integer c such that

\{a,a{ + }c,a{ + } 2 c,\ldots,a{ + }kc\}\subset A .

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]

Progress and related results

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. A 2016 paper by Bloom^[4] proved that if

A\subset \{ 1 ,..,N\} contains no non-trivial three-term arithmetic progressions then |A|\ll N( log { log {N}})/ log {N} .

In 2020 a preprint by Bloom and Sisask^[5] improved the bound to

|A|\ll N/( log {N}) ^{1+c} for some absolute constant c> 0 .

In 2023 a new bound of

\exp( - c( log N) ^{1/12} )N ^{[6] [7] [8]} was found by computers scientist Kelley an Meka and shortly after an exposition in more familiar mathematical terms was given by Bloom and Sisask^{[9] [10]} who have since also improved the exponent of the Kelly-Meka bound to \beta =1 / 9 (and conjectured \beta =5 / 41 ) in a preprint.^[11]

See also

• Problems involving arithmetic progressions
• List of sums of reciprocals
• List of conjectures by Paul Erdős
• Müntz–Szász theorem

References

• 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.

External links

• The Erdős–Turán conjecture or the Erdős conjecture? on MathOverflow

Notes and References

1. .
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.
4. 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 . 1405.5800 . 10.1112/jlms/jdw010 . 3509957 . 27536138.
5. Bloom . Thomas F. . Sisask . Olof . 2020 . Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions . math.NT . 2007.03528.
6. Kelley . Zander . Meka . Raghu . 2023-11-06 . Strong Bounds for 3-Progressions . 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)-Conference Proceedings . IEEE . 933–973 . 10.1109/FOCS57990.2023.00059 . 979-8-3503-1894-4. 2302.05537 .
7. 2302.05537 . math.NT . Zander . Kelley . Raghu . Meka . Strong Bounds for 3-Progressions . 2023-02-10.
8. Web site: Sloman . Leila . 2023-03-21 . Surprise Computer Science Proof Stuns Mathematicians . Quanta Magazine . en.
9. Bloom . Thomas F. . Sisask . Olof . 2023-12-31 . The Kelley–Meka bounds for sets free of three-term arithmetic progressions . Essential Number Theory . en . 2 . 1 . 15–44 . 10.2140/ent.2023.2.15 . 2834-4634. 2302.07211 .
10. 2302.07211 . math.NT . Thomas F. . Bloom . Olof . Sisask . The Kelley--Meka bounds for sets free of three-term arithmetic progressions . 2023-02-14.
11. 2309.02353 . math.NT . Thomas F. . Bloom . Olof . Sisask . An improvement to the Kelley-Meka bounds on three-term arithmetic progressions . 2023-09-05.