### Number Theory Problems Related to Toric Integer Orbits

Heilbronn Number Theory Seminar

12th June 2018, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

The affine sieve of Bourgain, Gamburd, and Sarnak allows one to prove the existence of

infinitely many integers having a bounded number of prime factors for all suitable polynomial

functions of integer orbits of thin discrete groups L in a linear algebraic group G having the

property that the Zariski closure of L is Levi-semisimple. The affine sieve results do not apply

to such groups whose Zariski closure has a nontrivial homomorphic image that is an algebraic

torus. We consider heuristics for integer orbits in algebraic tori where (conjecturally) where

only finitely many elements have any fixed number of prime factors. We present and analyse a

probabilistic model, refining a model of Salehi Golsefidy and Sarnak, which makes quantitative

predictions on growth rate of number of prime factors, and compare it to data. (This is joint

work with A. Kontorovich.)

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