טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentHoresh Tal
SubjectZariski Density of Prime Points in Orbits
DepartmentDepartment of Mathematics
Supervisor Professor Amos Nevo
Full Thesis textFull thesis text - English Version


Abstract

We use a variation on Vinogradov's three-prime theorem to prove Zariski density of prime points in several infinite families of affine algebraic varieties. Among the varieties considered are two infinite families of examples to a conjecture by J. Bourgain, A. Gamburd and P. Sarnak regarding prime points in orbits of simple groups. One of these two examples is the variety of matrices of fixed Pfaffian. In addition, we consider some non-homogeneous varieties, such as the variety of matrices of fixed permanent.

The variation of Vinogradov’s theorem is a theorem due to Vaughan, which does not have a published proof. We bring the details of the proof, which relies on the Hardy and Littlewood circle method.