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

We prove that if $A$ is any set of prime numbers satisfying $$\sum _{a\in A}\frac{1}{a}=\mathrm{\infty },$$ 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 $\alpha (N)=(\mathrm{\mid }A\mathrm{\mid }\log N)\mathrm{/}N$ at least $$\alpha (N){\gg }_B{(\log \log N)}^{-B}$$ then $A$ contains a $3$-term arithmetic progression.