Mathematics Colloquia and Seminars

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The affine sieve is a technique first developed by Bourgain, Gamburd, and Sarnak in 2006 and later completed by Salehi Golsefidy and Sarnak in 2010 which proves that in a broad class of affine linear actions, polynomial functions of the coordinates saturate. While this works in large generality, the bounds it produces on the saturation number are often far from optimal. Thin orbits of Pythagorean triangles have been of particular interest since the outset of the affine sieve, and I will discuss recent progress on improving bounds for saturation numbers of their hypotenuse, area, and the product of all three coordinates by using Archimedean sieve theory and the dispersion method.