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.