Three-term Progressions Research Articles

On the Ramsey number of the Brauer configuration

We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain non-linear quadratic equations.

Improved bounds for progression-free sets in $$C_8^{n}$$

Let G be a finite group, and let r3(G) represent the size of the largest subset of G without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that r3($$C_4^{n}$$) ≤ (3.611)n, where Cm denotes the cyclic group of order m. For finite abelian groups $$G \cong \prod\nolimits_{i = 1}^n {{C_{{m_i}}}} $$, where m1,…,mn denote positive integers such that m1 |…|mn, this also yields a bound of the form $$r_3(G)\leqslant(0.903)^{{rk}_4(G)}|G|$$, with rk4(G) representing the number of indices i ∈ {1,…, n} with 4 |mi. In particular, r3($$C_8^{n}$$) ≤ (7.222)n. In this paper, we provide an exponential improvement for this bound, namely r3($$C_8^{n}$$) ≤ (7.0899)n.

Szemerédi's Theorem in the Primes

AbstractGreen and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any subset of the primes of relative density tending to zero sufficiently slowly contains a three-term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of three-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.

Mixing for three-term progressions in finite simple groups

AbstractAnswering a question of Gowers, Tao proved that any A × B × C ⊂ SLd(𝔽q)3 contains |A||B||C|/|SLd(𝔽q)| + Od(|SLd(𝔽q)|2/qmin(d−1,2)/8) three-term progressions (x, xy, xy2). Using a modification of Tao's argument, we prove such a mixing result for three-term progressions in all nonabelian finite simple groups except for PSL2(𝔽q) with an error term that depends on the degree of quasirandomness of the group. This argument also gives an alternative proof of Tao's result when d > 2, but with the error term O(|SLd(𝔽q)|2/q(d−1)/24).