Green 2017 new bounds szemeredi s theorem

../

theorem_1_1: The largest subset of {1, ..., N} with no four-term arithmetic progression has size at most a constant times N(log N)^{-c}, for some absolute constant c > 0.

theorem_3_1: For every prime p, every 0 < eta <= 1/10 and every f from Z/pZ to [-1,1] there are random a and r, possibly dependent, with E f(a) near the mean of f, a four-term recurrence average at least (E f(a))^4 - O(eta), and r = 0 with small probability.

---

Green, Ben and Tao, Terence, New bounds for Szemerédi's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika 63 (2017), no. 3, 944-1040, doi:10.1112/S0025579317000316. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1705.01703), every other right reserved.

The copy read for this card is the arXiv version stamped "arXiv:1705.01703v3 [math.CO] 10 Aug 2017"; the theorem and page numbers cited here are that version's.

Theorem 1.1 (p. 2) shows r_4(N), the largest size of a subset of [N] with no four-term arithmetic progression, satisfies r_4(N) << N(log N)^{-c} for an absolute constant c > 0. This improves Gowers's r_4(N) << N(log log N)^{-c} and the authors' earlier r_4(N) << N exp(-c sqrt(log log N)), bringing the four-term bound to the same quality as the Heath-Brown-Szemeredi bound for r_3(N), and the authors say it appears to be the limit of their methods. Instead of Roth's density increment the proof uses an energy decrement and regularity approach inspired by the Khintchine-type recurrence theorems for length four progressions of Bergelson-Host-Kra (ergodic setting) and of the authors (combinatorial setting), working with Bohr sets, dilated tori, quadratic approximants, a dimension decrement when lower bounds fail, an energy decrement when approximation fails, and a local inverse U^3 theorem. The introduction records that Erdos's conjecture on sets whose reciprocals diverge is equivalent to sum_{n>=1} r_k(2^n)/2^n < infinity for all k >= 3, citing Tao and Vu's Additive combinatorics, Exercise 10.0.6. Theorem 1.1 is deduced (p. 6) from the recurrence theorem, Theorem 3.1 (p. 5), whose proof is the iteration of Proposition 3.3 (pp. 16-18) carried out in Sections 4-9.

Source: https://arxiv.org/abs/1705.01703.

Results.

• Theorem 1.1 (p. 2): r_4(N) << N(log N)^{-c} for some absolute constant c > 0.
• Theorem 3.1 (p. 5): for a prime p, 0 < eta <= 1/10 and f from Z/pZ to [-1,1], random a, r, not necessarily independent, with E f(a) within O(eta) of the mean of f, a four-term recurrence average at least (E f(a))^4 - O(eta), and a thickness bound on P(r = 0); the result page notes the sign of the exponent in that bound as printed.

Read status. Claims checked: Theorems 1.1 and 3.1 were read clause by clause on the printed pages of the version named above, with the deduction of Theorem 1.1 (p. 6) and of Theorem 3.1 from Proposition 3.3 (pp. 16-18); the rest of the proof was read for structure only.

Bears on. #139 (Theorem 1.1 gives r_4(N) = o(N), the instance k = 4, with a rate), #142 (Theorem 1.1 is an upper bound on r_4(N) only; no asymptotic formula and no lower bound), #3 (the introduction records the equivalence of the problem with sum_{n>=1} r_k(2^n)/2^n < infinity for all k >= 3; Theorem 1.1's bound sums over N = 2^n only when c > 1, which the theorem does not give, so it does not give the four-term case, and the paper does not claim it does)

Other results named here, without result pages.

• Theorem 8.1 (p. 51), the local inverse U^3 theorem, proved in Section 9: a 1-bounded function with large local U^3 norm on a Bohr set correlates, with polynomial bounds, with a locally quadratic phase on translates of a smaller, dilated Bohr set.
• Energy decrement scheme (Theorems 6.6 and 6.7, pp. 39-40): failure of approximation by a structured local approximant yields an energy decrement (Theorem 6.6, proved in Section 8 from Theorem 8.1), and a bad lower bound for the four-term count lowers by at least one the poorly distributed quadratic dimension, the largest torus dimension over the poorly distributed labels, while the rank of the Bohr sets may grow (Theorem 6.7, proved in Section 7).

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