Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Strong Bounds for 3-Progressions (2302.05537v6)

Published 10 Feb 2023 in math.NT and math.CO

Abstract: We show that for some constant $\beta > 0$, any subset $A$ of integers ${1,\ldots,N}$ of size at least $2{-O((\log N)\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic progressions were known to exist only for sets of size at least $N/(\log N){1 + c}$ for a constant $c > 0$. Our approach is first to develop new analytic techniques for addressing some related questions in the finite-field setting and then to apply some analogous variants of these same techniques, suitably adapted for the more complicated setting of integers.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (39)
  1. Improved bounds for the sunflower lemma. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 624–630, 2020.
  2. Felix A Behrend. On sets of integers which contain no three terms in arithmetical progression. Proceedings of the National Academy of Sciences, 32(12):331–332, 1946.
  3. New bounds on cap sets. Journal of the American Mathematical Society, 25(2):585–613, 2012.
  4. Parallel repetition for the GHZ game: Exponential decay. CoRR, abs/2211.13741, 2022.
  5. Thomas F Bloom. A quantitative improvement for Roth’s theorem on arithmetic progressions. Journal of the London Mathematical Society, 93(3):643–663, 2016.
  6. Jean Bourgain. On triples in arithmetic progression. Geometric and Functional Analysis, 9(5):968–984, 1999.
  7. Jean Bourgain. Roth’s theorem on progressions revisited. Journal d’Analyse Mathématique, 104(1):155–192, 2008.
  8. Logarithmic bounds for Roth’s theorem via almost-periodicity. DISCRETE ANALYSIS, 2019.
  9. Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions. arXiv preprint arXiv:2007.03528, 2020.
  10. Multi-party protocols. In David S. Johnson, Ronald Fagin, Michael L. Fredman, David Harel, Richard M. Karp, Nancy A. Lynch, Christos H. Papadimitriou, Ronald L. Rivest, Walter L. Ruzzo, and Joel I. Seiferas, editors, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 94–99. ACM, 1983.
  11. Mei-Chu Chang. A polynomial bound in Freiman’s theorem. Duke mathematical journal, 113(3):399–419, 2002.
  12. Progression-free sets in are exponentially small. Annals of Mathematics, pages 331–337, 2017.
  13. A probabilistic technique for finding almost-periods of convolutions. Geometric and functional analysis, 20(6):1367–1396, 2010.
  14. On large subsets of with no three-term arithmetic progression. Annals of Mathematics, pages 339–343, 2017.
  15. Michael Elkin. An improved construction of progression-free sets. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 886–905. SIAM, 2010.
  16. A tight bound for Green’s arithmetic triangle removal lemma in vector spaces. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1612–1617. SIAM, 2017.
  17. William T Gowers. A new proof of Szemerédi’s theorem. Geometric & Functional Analysis GAFA, 11(3):465–588, 2001.
  18. William T Gowers. What is difficult about the cap set problem? http://gowers.wordpress.com/2011/01/11/what-is-difficult-about-the-cap-set-problem/, 2011. Accessed: 1-5-2023.
  19. Ben Green. Progressions of length 3 following Szemerédi. https://people.maths.ox.ac.uk/greenbj/papers/szemeredi-roth.pdf, 1999.
  20. A note on Elkin’s improvement of Behrend’s construction. In Additive number theory, pages 141–144. Springer, 2010.
  21. David Rodney Heath-Brown. Integer sets containing no arithmetic progressions. Journal of the London Mathematical Society, 2(3):385–394, 1987.
  22. Structure of protocols for xor functions. SIAM Journal on Computing, 47(1):208–217, 2018.
  23. Mean, median and mode in binomial distributions. Statistica Neerlandica, 34(1):13–18, 1980.
  24. The growth rate of tri-colored sum-free sets. Discrete Analysis, page 3734, 2018.
  25. Shachar Lovett. An exposition of Sanders’ quasi-polynomial Freiman-Ruzsa theorem. Theory of Computing, pages 1–14, 2015.
  26. Roy Meshulam. On subsets of finite abelian groups with no 3-term arithmetic progressions. Journal of Combinatorial Theory, Series A, 71(1):168–172, 1995.
  27. Kevin O’Bryant. Sets of integers that do not contain long arithmetic progressions. the electronic journal of combinatorics, 18(P59):1, 2011.
  28. Klaus F Roth. On certain sets of integers. J. London Math. Soc, 28(104-109):3, 1953.
  29. Tom Sanders. Popular difference sets. Online J. Anal. Comb., 5(5):4, 2010.
  30. Tom Sanders. On Roth’s theorem on progressions. Annals of Mathematics, pages 619–636, 2011.
  31. Tom Sanders. On certain other sets of integers. Journal d’Analyse Mathématique, 1(116):53–82, 2012.
  32. Tom Sanders. On the Bogolyubov–Ruzsa lemma. Analysis & PDE, 5(3):627–655, 2012.
  33. Tomasz Schoen. New bounds in Balog-Szemerédi-Gowers theorem. Combinatorica, 35(6):695–701, 2015.
  34. Tomasz Schoen. Improved bound in Roth’s theorem on arithmetic progressions. Advances in Mathematics, 386:107801, 2021.
  35. ID Shkredov. Some remarks on the Balog–Wooley decomposition theorem and quantities D+, D×\times×. Proceedings of the Steklov Institute of Mathematics, 298:74–90, 2017.
  36. Higher moments of convolutions. Journal of Number Theory, 133(5):1693–1737, 2013.
  37. Roth’s theorem for four variables and additive structures in sums of sparse sets. In Forum of Mathematics, Sigma, volume 4. Cambridge University Press, 2016.
  38. Endre Szemerédi. Integer sets containing no arithmetic progressions. Acta Mathematica Hungarica, 56(1):155–158, 1990.
  39. Additive combinatorics, volume 105. Cambridge University Press, 2006.
Citations (40)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now