  We show that if besides the primes some other sequences (involving the
Liouville function and the primes) have a common distribution level exceeding
0.7231 then for any positive even integer $h$ there are arbitrarily long
arithmetic progressions of primes $p$ such that $p+h$ is also prime for each
element of the progression. In case of $h=2$ this means that under some
plausible unproved hypotheses about regular distribution of the primes and
other sequences in arithmetic progressions we really have arbitrarily long
arithmetic progressions in the sequence of twin primes.
