the affine sieve is a technique first developed by bourgain, gamburd, and sarnak in 2006 and later completed by salehi golsefidy and sarnak in 2010 to study almost-primality in a broad class of affine linear actions. the beauty of this is that it gives us effective bounds on the saturation number for thin orbits coming from $gl_n$ - in particular, producing infinitely many $r$-almost primes for some $r$. however, in practice this value of $r$ is often far from optimal. the case of thin pythagorean triangles has been of particular interest since the outset of the affine sieve, and i will discuss recent progress on improving bounds for the saturation numbers for their hypotenuses and areas using archimedean sieve theory.