A Density Increment Approach to Roth's Theorem in the Primes

Abstract

We prove that if $A$ is any set of prime numbers satisfying $$\sum_{a\in A}\frac{1}{a}=\infty,$$ then $A$ must contain a $3$-term arithmetic progression. This is accomplished by combining the transference principle with a density increment argument, exploiting the structure of the primes to obtain a large density increase at each step of the iteration. The argument shows that for any $B>0$, and $N>N_{0}(B)$, if $A$ is a subset of primes contained in $\{1,\dots,N\}$ with relative density $\alpha(N)=(|A|\log N)/N$ at least $$\alpha(N)\gg_{B}\left(\log\log N\right)^{-B}$$ then $A$ contains a $3$-term arithmetic progression.Comment: This has paper has been withdrawn due to an error in equation (2.8). This error comes from the linearization step. I believe that the density increment argument can be corrected, and a similar bound can be obtained by moving entirely to Bohr sets. Currently this paper has a hole so I am removing it from the arXi