  We show that using character sum estimates due to H. Iwaniec leads to an
improvement of recent results about the distribution and finding RSA moduli
$M=pl$, where $p$ and $l$ are primes, with prescribed bit patterns. We are now
able to specify about $n$ bits instead of about $n/2$ bits as in the previous
work. We also show that the same result of H. Iwaniec can be used to obtain an
unconditional version of a combinatorial result of W. de Launey and D. Gordon
that was originally derived under the Extended Riemann Hypothesis.
