Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions

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corollary_1_2: Bloom and Sisask's corollary that a set of natural numbers whose reciprocals sum to infinity contains infinitely many non-trivial three-term arithmetic progressions, the case of length three of Erdős's conjecture on arithmetic progressions.

corollary_1_3: Bloom and Sisask's bound for primes: a subset of the primes up to N with no non-trivial three-term arithmetic progression has relative density in those primes at most a constant times 1/(log N)^c, for an absolute constant c > 0.

theorem_1_1: Bloom and Sisask's bound in Roth's theorem: a subset of {1,...,N}, N at least 2, with no non-trivial three-term arithmetic progression has at most a constant times N/(log N)^(1+c) elements, for an absolute constant c > 0.

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Thomas F. Bloom, Olof Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. arXiv:2007.03528 (2020).

The copy read for this card is the arXiv preprint, arXiv:2007.03528v2 (1 September 2021).

Theorem 1.1 (p. 1) proves that if $N\ge2$ and $A\subset\{1,\ldots,N\}$ has no non-trivial three-term arithmetic progression (no solution of $x+y=2z$ with $x\neq y$), then $|A|\ll N/(\log N)^{1+c}$ for an absolute constant $c>0$. This pushes past the $N/\log N$ density barrier: the previous best bound was $N/(\log N)^{1-o(1)}$, with proofs by Sanders, Bloom, Bloom and Sisask, and Schoen. Corollary 1.2 (p. 2) deduces, by partial summation, the first non-trivial case of Erdős's conjecture on arithmetic progressions: if the reciprocals of $A\subset\mathbb N$ sum to infinity then $A$ has infinitely many non-trivial three-term progressions. Corollary 1.3 (p. 2) bounds the relative density of a three-term-progression-free set of primes up to $N$ by $\ll1/(\log N)^c$. The proof is a density-increment argument over Bohr sets, using additive frameworks, additive structure of spectra and symmetry sets, a structure theorem for non-smoothing sets, and a spectral boosting step (Sections 4-12, pp. 17-91). The constant $c$ is in principle effectively computable, but the authors state that any value the method produces would be very small. Section 13 (pp. 91-94) conjectures (Conjecture 13.1, p. 91) that the largest density $r(N)$ of a three-term-progression-free subset of $\{1,\ldots,N\}$ satisfies $r(N)\ll\exp(-c'(\log N)^c)$ for all sufficiently large $N$, for some absolute constants $c,c'>0$, and poses Conjectures 13.2-13.5 on the additive structure of spectra.

Read status: claims checked for Theorem 1.1 and Corollaries 1.2 and 1.3 (pp. 1-2), the final assembly in Section 12 (pp. 89-91) and the conjectures of Section 13 (pp. 91-94), read on the printed pages; the deduction of Corollary 1.2 was followed step by step. Sections 4-11 were not checked. The result pages record each reading.

Source: https://arxiv.org/abs/2007.03528. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2007.03528), every other right reserved.

Bears on. #3: Corollary 1.2 (p. 2) proves the problem's statement for progressions of length three, in the form that a set with divergent reciprocal sum contains infinitely many non-trivial three-term progressions, deduced from Theorem 1.1; the paper calls it the first non-trivial case of the conjecture and does not treat longer progressions.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.