Improving Roth's Theorem in the Primes

Abstract

Abstract. Let A be a subset of the primes. Let δP (N ) = |{n ∈ A : n ≤ N }| |{n prime : n ≤ N }| . We prove that, if δP (N ) ≥ C log log log N (log log N ) 1/3 for N ≥ N0, where C and N0 are absolute constants, then A ∩ [1, N ] contains a non-trivial three-term arithmetic progression.This improves on Green's result [Gr], which needs δP (N ) ≥ C ′ s log log log log log N log log log log N .

On Systems of Complexity One in the Primes

“…Note that simply applying [13, Theorem D.3] would be insufficient for our purpose, since the error there is e O( √ ω) (log N) −1/20 and therefore it is non-trivial only for ω c(log log N) 2 , thus rendering the methods of Helfgott and de Roton [14] unapplicable. The argument of [12] also requires a modulus ω c log log N. Our construction follows closely that in [13, Appendix D], however with one important difference: we make a stronger assumption of finite complexity on the system of linear forms, and under this assumption we obtain improved estimates on the Euler products involved.…”

“…It is then a well-known result of Green [9] that every subset of P N of positive density contains a non-trivial three-term arithmetic progression; and the extension of this result to progressions of any length is the celebrated Green-Tao theorem [12]. Green's argument [9] actually allowed for densities as low as (log log log log N) −1/2+o (1) , and Helfgott and de Roton [14] later obtained a remarkable quantitative strenghtening of this result. Theorem 1 (Helfgott, de Roton).…”

“…To motivate Theorem 2, we now give some illustrative examples of systems of complexity one. First, any single translation-invariant equation has complexity one, although in that case a simple modification of the argument of Helfgott and de Roton [14] yields Theorem 2. A more representative example of a system of complexity one is that of "d points and their midpoints", corresponding to the set of equations (y ii + y jj = 2y ij ) 1 i<j d , whose solutions over Q are parametrized, with some multiplicity, by 4 ψ(x) = (x 0 + x i + x j ) 1 i j d .…”

“…Denoting by λ A the renormalized indicator function of a dense subset A of the primes, we therefore compare the average of λ A over ψ-patterns to that of a smoothed version λ ′ A of itself, which behaves as a dense subset of the integers of almost the same density. As usual, there is a little technical subtelty in the form of the W -trick, by which we consider, instead of the set A, its intersection with an arithmetic progression of modulus W = p ω p. A critical feature of Helfgott and de Roton's argument [14] is then that it requires a modulus ω ∼ c log N. At this point we invoke a beautiful recent result of Shao [26], who improved on a first result of Dousse [3], and generalized the logarithmic bounds of Bourgain [1] for Roth's theorem to a model system of complexity one. More precisely, Shao [26] investigated the system ψ(x) = (x 0 + x i + x j ) 1 i j d , and proved that a set A of density (log N) −1/6d(d+1)+o (1) in [N] contains a non-trivial configuration ψ(x) ∈ A d(d+1)/2 .…”

An -Function-Free Proof of Vinogradov’s Three Primes Theorem

“…In the application to Roth's theorem in the primes, this causes an extra layer of logarithm in the lower bound for the density threshold. However, this extra layer of logarithm was removed by Helfgott and de Roton [15] (whose result is further improved by Naslund [21,22]). Such an improvement comes from using a weaker L 2 estimate instead of an L ∞ estimate, but at the cost of decreasing X.…”