Almost-prime times in horospherical flows on the space of lattices

Abstract

We study the asymptotic distribution of almost-prime entries of abelian horospherical flows on Γ\SL n (R), where Γ is either SL n (Z) or a cocompact lattice. In the cocompact case, we obtain a result that implies density of almost-primes of a sufficient order, and in the space of lattices we show the density of almost-primes in the 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 sequences of times in abelian horospherical flows.

Polynomial effective density in quotients of $\mathbb H^3$ and $\mathbb H^2\times\mathbb H^2$

“…In this case, the behaviour of individual orbits can be related to decay of matrix coefficients, and hence effective equidistribution with polynomial error rate can be established. The first works in this direction we are aware of are [50,10,33] as well as the more recent [25,56,51] and this has now been established in much greater generality [32,43,31]. Closely related is the case of translates of periodic orbits of subgroups L ⊂ G which are fixed by an involution [15,22,2].…”

Quantitative disjointness of nilflows from horospherical flows

“…Recent work recovering a similar result (in a greater generality with respect to function spaces) was done by McAdam [31],…”

“…We end this section by proving an effective discrete version of Theorem 1.6, as such results are of interest in some applications and moreover, the idea of using the disjointness as a method to apply certain summation process by studying approperiate spectral kernels have been used by most the papers studying sparse equidistribution up to date [41,9,40,31] and will be used in subsequent paper of the author towards applications in sparse equdistribution problems [19]. We prove the theorem only in the case of abelian horospherical group, where the samplings from H are drawn along an abelian subgroup isomorphic to Z dim H , by using Venkatesh's method.…”

A sparse equidistribution result for $(\operatorname {SL}(2,\mathbb {R})/\Gamma _0)^{n}$

“…Apart from works of Shah [16], Venkatesh [21], Tanis and the author [19] and Flaminio, Forni and Tanis [4], there haven't been many results available which establish such sparse equidistribution results for every such orbit. Some of the newer results in similar directions include those of Katz [10] (on effective sparse equidistribution of horocycle orbits at times lying inside annuli) and that of McAdam [14] (on horocycle flow at almost prime times).…”