This project is in additive combinatorics - a field that lies between combinatorics, number theory, and harmonic analysis (in this project we will not see any harmonic analysis). 

 

Consider the polynomial f(x,y,z) and a finite set A of integers, we write

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f(A,A,A)={f(a,b,c) : a,b,c ∈ A}.

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In other words, f(A,A,A) is the set of all values that we obtain by plugging elements of A into f(x,y,z). We say that f is an expander if |f(A,A,A)| is much larger than |A|, for every set A that is not small.

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When f(x,y,z)=x+yz, it is know that |f(A,A,A)|≥|A|     (when A is not small). We thus say that x+yz is an expander. It is possible that the correct lower bound is much larger than |A|   .

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When f(x,y,z)=x+y+z and A is an arithmetic progression, |f(A,A,A)|<3|A|. We thus say that x+y+z is not an expander.

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The study of expander polynomials is much wider. It involves polynomials with different numbers of variables, over other fields, and much more. These problems have applications in combinatorics, theoretical computer science, number theory, and model theory.  In this project, we will explore a variety of such questions.

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For additional reading, see chapter 2 of Adam's additive combinatorics lecture notes. (It might be helpful to start with the first two pages of chapter 1, but not more than that.)