##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Max Ehrman

#### Yale University

## Almost prime coordinates in thin Pythagorean triangles

##### Abstract:

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 to study almost-primality in a broad class of affine linear actions. The beauty of this is that it gives us effective bounds on the saturation number for thin orbits coming from $GL_n$ - in particular, producing infinitely many $R$-almost primes for some $R$. However, in practice this value of $R$ is often far from optimal. The case of thin Pythagorean triangles has been of particular interest since the outset of the affine sieve, and I will discuss recent progress on improving bounds for the saturation numbers for their hypotenuses and areas using Archimedean sieve theory.

Host: Alireza Salehi Golsefidy

### November 14, 2016

### 2:00 PM

### AP&M 7321

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