  The celebrated Green-Tao theorem states that there are arbitrarily long
arithmetic progressions in the primes. One of the main ingredients in their
proof is a relative Szemer\'edi theorem which says that any subset of a
pseudorandom set of integers of positive relative density contains long
arithmetic progressions.
  In this paper, we give a simple proof of a strengthening of the relative
Szemer\'edi theorem, showing that a much weaker pseudorandomness condition is
sufficient. Our strengthened version can be applied to give the first relative
Szemer\'edi theorem for $k$-term arithmetic progressions in pseudorandom
subsets of $\mathbb{Z}_N$ of density $N^{-c_k}$.
  The key component in our proof is an extension of the regularity method to
sparse pseudorandom hypergraphs, which we believe to be interesting in its own
right. From this we derive a relative extension of the hypergraph removal
lemma. This is a strengthening of an earlier theorem used by Tao in his proof
that the Gaussian primes contain arbitrarily shaped constellations and, by
standard arguments, allows us to deduce the relative Szemer\'edi theorem.
