  Green and Tao famously proved in 2005 that any subset of the primes of fixed
positive density contains arbitrarily long arithmetic progressions. Green had
previously shown that in fact any subset of the primes of relative density
tending to zero sufficiently slowly contains a 3-term term progression. This
was followed by work of Helfgott and de Roton, and Naslund, who improved the
bounds on the relative density in the case of 3-term progressions. The aim of
this note is to present an analogous result for longer progressions by
combining a quantified version of the relative Szemer\'edi theorem given by
Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.
