there is a rich connection between homogeneous dynamics and number theory, especially when dynamical results are effective (i.e. when rates of convergence for dynamical phenomena are known). in this final defense, i describe my research on the asymptotic distribution of almost-prime times in horospherical flows on the space of lattices, as well as on compact quotients of sl(n,r). in the compact setting, i obtain a result that implies density for almost-primes in horospherical flows, where the number of prime factors is independent of the basepoint, and in the space of lattices i show the density of almost-primes in abelian horospherical orbits of points satisfying a certain diophantine condition. to prove this, i first give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, which i then use to prove an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows, to which i then apply a combinatorial sieve.