On Improving Roth's Theorem in the Primes

Abstract

Let $A\subset\left\{ 1,\dots,N\right\} $ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density $\alpha=|A|/\pi(N)$, where $\pi(N)$ denotes the number of primes in the set $\left\{ 1,\dots,N\right\} $. By modifying Helfgott and De Roton's work, we improve their bound and show that $$\alpha\ll\frac{\left(\log\log\log N\right)^{6}}{\log\log N}.$$