Almost-primes in horospherical flows on the space of lattices

TL;DR

This paper investigates the distribution of almost-prime entries in abelian horospherical flows on the space of lattices, providing new density and equidistribution results with effective rates.

Contribution

It introduces new density results for almost-primes in horospherical flows and offers effective equidistribution theorems with explicit rates.

Findings

Density of almost-primes in cocompact horospherical flows.

Density of almost-primes in orbits satisfying Diophantine conditions.

Effective equidistribution results with explicit rates.

Abstract

We study the asymptotic distribution of almost-prime entries of abelian horospherical flows on {\Gamma}\SL(n,R), where {\Gamma} 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.

Full text

Introduction

Preliminaries

Equidistribution

Arithmetic

Sieving

Conclusion

Appendices

Acknowledgements

I would like to thank my advisor, Amir Mohammadi, for giving me the idea to work on this problem and for providing important guidance throughout the process. I also want to thank Manfred Einsiedler and Hee Oh for helpful discussions about this work.