On certain other sets of integers

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We show that if A ⊂ {1,...,N} contains no non-trivial three-term arithmetic progressions then |A| = O(N/log3/4−o(1)N).

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Higher Order Fourier Analysis (2012, result): “Even more recently, this argument was combined with the Szemerédi-Heath-Brown argument by Bourgain [Bo2008], and refined further by Sanders [Sa2010], to obtain the further improvement of N0 exp(O(δ−4/3−o(1))), and then (by a somewhat different argument of Sanders [Sa2010]) of N0 exp(O(δ−1−o(1))).”
On Roth's theorem on progressions (2010, methodology): “dissociated (if µ = µG and K = 0 it is precisely this), but the reader does not need to have a deep understanding for the purposes of this paper as it is only used to couple the next two results from [San10] into Lemma 2.8. Lemma 2.5 (The Chang bound, [San10, Lemma 4.6]). Suppose that 0 ≡ f ∈ L2(µ).”
Surveys in Combinatorics 2011: Hypergraph Turán problems (2011, background): “It would be very interesting to tighten these bounds: this is connected with regularity theory (see Section 10) and bounds for Roth’s theorem (see [90, 166].”
Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. (2020, unknown): “It is present (although not in this language) in the work of Sanders [20, Lemma 6.2], the first author [4, Theorem 7.1], and the authors [5, Proposition 5.7].”