Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits
  We make an observation which doubles the exponent of distribution in certain Affine Sieve problems, such as those considered by Liu-Sarnak, Kontorovich, and Kontorovich-Oh. As a consequence, we decrease the known bounds on the saturation numbers in these problems. 
Jiuzu Hong and Alex Kontorovich
math.NT