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We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$.
On sets of integers which contain no three terms in arithmetical progression
Behrend, F. A · 1946
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On certain sets of integers
Roth, K. F · 1953
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Integer sets containing no arithmetic progressions
Heath-Brown, D. R · 1987
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Integer sets containing no arithmetic progressions
Szemerédi, E · 1990
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On subsets of finite abelian groups with no 3 3 -term arithmetic progressions
Meshulam, R · 1995
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On triples in arithmetic progression
Bourgain, J · 1999
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A polynomial bound in Freiman’s theorem
Chang, M.-C · 2002
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An improved construction of progression-free sets
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Bloom, T. F · 2014
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On improving Roth’s theorem in the primes
Naslund, E · 2015
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New bounds in Balog-Szemerédi-Gowers theorem
Schoen, T · 2015
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A quantitative improvement for Roth’s theorem on arithmetic progressions
Bloom, T. F · 2016
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Roth’s theorem for four variables and additive structures in sums of sparse sets
Schoen, T., and Sisask, O · 2016
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Progression-free sets in ℤ 4 n \mathbb{Z}^{n}_{4} are exponentially small
Croot, E., Lev, V. F., and Pach, P. P · 2017
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Sanders, T · 2011
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New bounds on cap sets
Bateman, M., and Katz, N. H · 2012
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On certain other sets of integers
Sanders, T · 2012
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On large subsets of 𝔽 q n \mathbb{F}^{n}_{q} with no three-term arithmetic progression
Ellenberg, J. S., and Gijswijt, D · 2017
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Logarithmic bounds for Roth’s theorem via almost-periodicity
Bloom, T. F., and Sisask, O · 2019
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Improved bound in Roth’s theorem on arithmetic progressions, 2020
Schoen, T · 2020
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