Effective Equidistribution in Homogeneous Dynamics with Applications in Number Theory

We study the asymptotic distribution of almost-prime entries in horospherical flows on the quotient of SL(n,R) by a lattice, where the lattice is either cocompact or SL(n,Z). In the cocompact case, we 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 we show the density of almost-primes in abelian horospherical orbits of points satisfying a certain Diophantine condition. Along the way we give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, as well as an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows