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

Taylor McAdam

doi:10.3934/jmd.2019022

Article Contents

2019, Volume 15: 277-327. Doi: 10.3934/jmd.2019022

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

Taylor McAdam

Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, CT 06520-8283, USA

Received: June 05, 2018

Revised: April 19, 2019

Published: November 2019

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Abstract

An integer is called almost-prime if it has fewer than a fixed number of prime factors. In this paper, we study the asymptotic distribution of almost-prime entries in horospherical flows on $ \Gamma\backslash {{\rm{SL}}}_n(\mathbb{R}) $, where $ \Gamma $ is either $ {{\rm{SL}}}_n(\mathbb{Z}) $ or a cocompact lattice. 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 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.

Keywords:

Mathematics Subject Classification: Primary: 37D40; Secondary: 11N05.

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Figure 1. The symmetric difference between $ B_T $ and $ B_T - (H,\ldots,H) $

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Figure 2. The measure of the set where $ |{\bf s_1-s_2| }<\epsilon $ has measure bounded by $ H^d\epsilon^d $ in $ B_H\times B_H $ (shown here for one dimensional $ U $)

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Figure 3. The area shaded in solid gray indicates the region over which we are integrating in the definition of $ S_{\rm approx} $, whereas the area shaded with diagonal lines represents the region over which we are integrating in our estimate of $ S_{\rm approx} $ given in (43). The difference between the two integrals can be bounded by the number of $ \delta $-cubes intersecting the boundary of $ B_T $ multiplied by the supremum of $ f $

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Figure 4. In $ S_K(A) $ we are summing over the integer points in $ \tilde B_T $ such that $ K|k_1\cdots k_2 $ (filled in black). We may do this by summing over shifted grids based at each of the points in the first box $ \tilde B_K $ (filled in gray). However, this introduces an error determined by $ \mathscr{S}_{{\infty},{0}}(f) $ and the number of points in each of these shifted grids falling outside $ B_T $ (filled in white). The number of such points can be bounded by $ T^{d-1}K^{1-d} $, as we have seen before

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