  We study an instance of the Affine Sieve, producing a level of distribution
beyond that which can be obtained from current techniques, even assuming a
Selberg/Ramanujan-type spectral gap. In particular, we consider the set of
hypotenuses in a thin orbit of Pythagorean triples. Previous work [Kon07,
Kon09, KO12] gave an exponent of distribution alpha < 1/12 coming from
Gamburd's [Gam02] gap theta = 5/6, thereby producing R = 13 almost primes in
this linear sieve problem (see Sec. 1 for definitions). If conditioned on a
best possible gap theta = 1/2, the known method would give an exponent alpha <
1/4, and R = 5 almost primes. The exponent 1/4 is the natural analogue of the
"Bombieri-Vinogradov" range of distribution for this problem, see Remark 1.19.
  In this paper, we unconditionally prove the exponent alpha < 7/24 (in the
"Elliott-Halberstam" range), thereby producing R = 4 almost primes. The main
tools involve developing bilinear forms and the dispersion method in the range
of incomplete sums for this Affine Sieve problem.
