A Subexponential Upper Bound for van der Waerden Numbers $W(3,k)$

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$


Introduction
Let k and l be positive integers. The van der Waerden number W (k, l) is the smallest positive integer N such that in any partition {1, . . . , N } = X ∪ Y there is an arithmetic progression of length k in X or an arithmetic progression of length l in Y . The existence of such numbers was established by van der Waerden [20], however the order of magnitude of W (k, l) is unknown for k, l 3. Clearly, W (k, l) is related to Szemerédi's theorem on arithmetic progressions [18] and any effective estimate in this theorem leads to an upper bound on the van der Waerden numbers. Currently best known bounds in the most important diagonal case are The upper bound follows from the famous work of Gowers [11] and the lower bound was proved by Szabó [17] using probabilistic argument. Furthermore, Berlekamp [3] showed that if k − 1 is a prime number then Another very intriguing instance the problem are numbers W (3, k) as they are related to Roth's theorem [14] that provides more efficient estimates for sets avoiding three-terms arithmetic progressions. Let us denote by r(N ) the size of the largest progression-free subset of {1, . . . , N }. We know that see [4,5,15,16]. However this bound is not strong enough to imply a subexponential estimate for W (3, k).
Green [12] proposed a very clever argument based on arithmetic properties of sumsets to bound W (3, k). Building on this method and applying results from [9] it was showed in [10] that The best known lower bound was obtained by Li and Shu [13] (see also [7]), who showed that The purpose of this paper is to prove a subexponential bound on W (3, k).
Theorem 1 There are absolute constants C, c > 0 such that for every k we have Our argument is based on the method of [16], which explores in details the structure of a large spectrum. This method can be partly applied (see Lemma 5) in our approach and it deals only with a progression-free partition class. The second part of the proof exploits the structure of both partition classes and in this case the argument of [16] has to be significantly modified.

Notation
The Fourier coefficients of a function f : Z/N Z → C are defined by where r ∈ Z/N Z. The inversion formula states that We denote by 1 A (x) the indicator function of set A. Thus using the inversion formula and the fact that (1 A * 1 B )(r) = 1 A (r) 1 B (r) one can express the number of three-term arithmetic progressions (including trivial ones) by Parseval's identity asserts in particular that Let θ 0 be a real number, the θ−spectrum of A is defined by If A is specified then we write ∆ θ instead of ∆ θ (A). By the span of a finite set S we mean Chang's Spectral Lemma provides an upper bound for the dimension of a spectrum.
We are going to use Bohr sets [6] to prove the main result. Let Γ ⊆ G and γ ∈ (0, 1 2 ] then the Bohr set generated by Γ with radius γ is where x = min y∈Z |x − y|. The rank of B is the size of Γ and we denote it by rk(B). Given η > 0 and a Bohr set B = B(Γ, γ), by B η we mean the Bohr set B(Γ, ηγ). We will use two basic properties of Bohr sets concerning its size and regularity, see [19].

Proof of Theorem 1
Our main tool is the next lemma, which can be extracted from [16] (see Lemmas 7,9,12 and 13). Its proof makes use of the deep result by Bateman and Katz in [1,2] describing the structure of the large spectrum.
Lemma 5 [16] There exists an absolute constant c > 0 such that the following holds. Let A ⊆ Z/N Z, |A| = δN be a set without any non-trivial arithmetic progressions of length three and such that Then there is a regular Bohr set B with rk(B) ≪ δ −1+c and radius Ω(δ 1−c ) such that for some t Furthermore, we apply Bloom's iterative lemma, that provides a density increment by a constant factor greater than 1 for progression-free sets and Sanders' lemma on a containment of long arithmetic progressions in dense subsets of regular Bohr sets.