On Improving Roth's Theorem in the Primes

Abstract

Let A ⊂ {1, . . . , N } be a set of prime numbers containing no nontrivial arithmetic progressions. Suppose that A has relative density α = |A|/π(N ), where π(N ) denotes the number of primes in the set {1, . . . , N }. By modifying Helfgott and De Roton's work [Improving Roth's theorem in the primes. Int. Math. Res. Not. IMRN 2011 (4) (2011), 767-783], we improve their bound and show thatIn 1939, Van Der Corput [13] showed that P contains infinitely many nontrivial three-term arithmetic progressions. Green [4] proved an analogue of

Szemerédi's Theorem in the Primes

“…We do not see any fundamental obstruction to extending the result in this paper to more general systems of translation-invariant linear equations for which a result of Varnadives strength is known, but shall not attempt to do so here. It would arguably be of greater interest to find a more direct approach for longer progressions along the lines of [10] and [12], where the bounded function f is constructed explicitly.…”

“…, N } not containing any non-trivial k-term arithmetic progressions, and let r k (P N ) denote the maximal size of a subset of the set P N := P ∩ [N ] of primes less than N not containing any non-trivial k-term arithmetic progressions. Define the corresponding critical densities by Regarding relative density in the primes, the current record for progressions of length 3 is α 3 (P N ) ≪ (log log N ) −1+o(1) , arrived at through a series of articles by Green [Gre05], Helfgott and de Roton [HdR11], and finally Naslund [Nas15]. Henriot [Hen16] extended Naslund's result to all linear systems of complexity one, and our result relies crucially on the optimised estimates of the enveloping sieve weights he gave in this paper.…”

On Systems of Complexity One in the Primes

“…, λ ′ A ), we invoke a key transference estimate of Helfgott and de Roton [14], which essentially allows us to consider λ ′ A as a subset of the integers of density α 2 . It is further possible, by a result of Naslund 7 [20], to obtain an exponent 1 + o(1) instead of 2, and we choose to work with that more efficient version, even though it is possible to derive Theorem 2 with a smaller exponent without it. This is because we wish to exhibit that our argument preserves the exponent in Szemerédi-type theorems in the integers, in the sense of Proposition 5 below.…”

“…The main structure of our argument follows the transference principle, introduced by Green [11] and further developed by Green and Tao [14], and by which one lifts a dense subset of the primes to a dense subset of the integers. More precisely, we initially follow the transference strategy of Helfgott and de Roton [16], which builds on that of Green and Tao [12], and we incorporate Naslund's [22] estimates. 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.…”

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.…”