The dichotomy between structure and randomness , arithmetic progressions , and the primes

  title={The dichotomy between structure and randomness , arithmetic progressions , and the primes},
  author={Terence Tao},
A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of… CONTINUE READING
30 Citations
65 References
Similar Papers


Publications referenced by this paper.
Showing 1-10 of 65 references

Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions

  • H. Furstenberg
  • J. Analyse Math
  • 1977
Highly Influential
10 Excerpts

Über Summen von Primzahlen und Primzahlquadraten

  • J. G. van der Corput
  • Math. Ann
  • 1939
Highly Influential
4 Excerpts

On certain sets of positive density

  • P. Varnavides
  • J. London Math. Soc
  • 1959
Highly Influential
3 Excerpts

On certain sets of integers

  • K. F. Roth
  • J. London Math. Soc
  • 1953
Highly Influential
7 Excerpts

Arithmetic progressions in the primes

  • T. Tao
  • Collect. Math. (2006),
  • 2006
Highly Influential
5 Excerpts

A mean ergodic theorem for 1/N ∑N n=1 f (T x)g(T n 2 x)

  • H. Furstenberg, B. Weiss
  • Ohio State Univ. Math. Res. Inst. Publ
  • 1996
Highly Influential
4 Excerpts

Similar Papers

Loading similar papers…