Basepoint-independent density of almost-primes in horospherical orbits in SL(3,Z)\SL(3,R)

Group Actions and Dynamics
Event time: 
Monday, April 26, 2021 - 4:00pm
Taylor McAdam
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Event description: 

 Inspired by the work of Sarnak and Ubis [1] in SL(2,Z)\SL(2,R), we prove that almost-prime times (i.e. integer times having fewer than a fixed number of prime factors) in horospherical orbits of generic points in SL(3,Z)\SL(3,R) are dense in the whole space, where the number of prime factors allowed in the almost-primes is independent of the basepoint.  This is in contrast to previous work [2] in which the number of prime factors depends on a Diophantine property of the basepoint.  The proof involves a case-by-case analysis of the different ways in which a basepoint can fail the Diophantine property.  If a basepoint fails to equidistribute rapidly in the whole space with respect to the continuous time flow, then there exists a sequence of nearby periodic orbits of increasing volume that approximate the original orbit up to larger and larger time scales, and which equidistribute in the whole space as the volume grows. Given an open set, one can find a large enough periodic orbit such that almost-primes of a fixed order in the periodic orbit land inside that set, and this property can then be transported to the nearby orbit of the original basepoint.  This is joint work-in-progress with Manuel Luethi.

[1] Sarnak, Peter, and Adrián Ubis. “The horocycle flow at prime times.” Journal de mathématiques pures et appliquées 103.2 (2015): 575-618.
[2] McAdam, Taylor. “Almost-prime times in horospherical flows on the space of lattices.” Journal of Modern Dynamics 15 (2019): 277.