Professor Ernie Croot, who has just started as tenure track professor at Georgia Tech, will offer a Special Topics course in "Arithmetic Combinatorics." This course listed as a Graduate and a Undergraduate course.
Math 4803: Arithmetic Combinatorics MWF 1105-1155am
Arithmetic combinatorics is a branch of number theory and combinatorics that aims to show that every subset of the integers with certain
prescribed properties must have a particular substructure. For example,
a classic theorem due to K. F. Roth asserts that for all sufficiently large integers X, any subset S of {1,2,...,x} having at least x/\log\log x members must contain a three-term arithmetic progression. By this we mean a must contain a three-term arithmetic progression, by which we mean a
triple of integers a,b,c, with a and b distinct, such that a+b = 2c.
The main results in this area are proved using a variety of methods and techniques taken from graph theory, probability theory, geometric
combinatorics, harmonic analysis, and analytic number theory (in particular, the circle method). Even so, no prior background, except a knowledgeof elementary number theory, basic analysis, combinatorics, and
probabilitytheory, will be necessary. The main requirement will be mathematical maturity, and an ability to read through and present proofs.
The main topics I plan to cover are as follows:
1. Schur's theorem on monochromatic solutions to $x+y=z$, basic
Ramsey Theory, and van der Waerden's theorem.
2. The classical combinatorial inequalities: Cauchy-Davenport
inequality,
Shnirel'man's theorem, Kneser's inequality, and others.
3. Basic estimates for points on varieties on ${\bf Z}/N{\bf Z}$ and
$F_q$ (where $q$ is a prime power). The methods used here will be
purely
classical, and will mostly involve Gauss and Jacobi sums.
4. The circle method and applications to Waring's problem.
5. Behrend's lower bound for the densest subsets of the integers free
of 3-term arithmetic progressions. Rankin-Laba-Lacey's result for
k-term progressions.
6. Roth's theorem on 3-term arithmetic progressions. Sarkozy's result.
7. Szemeredi's theorem, and H. Furstenberg's ergodic theory proof.
8. Szemeredi and Heath-Brown's Improvement of Roth's theorem.
9. Plunnecke's inequality, Ruzsa's finite group version of Freiman's
theorem, and the version in the integers.
10. Lower bounds for the maximum size of the product set $A \cdot A$
and sumset $A+A$: The Szemeredi-Trotter theorem, and Elekes's
application
to sums and products.
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Depending on the amount of time it takes to get through these topics, I
may decide to remove some from the list, but I may also decide to add
a few more. In addition to my own weekly lectures, I will try to
encourage
everyone to read through some of the papers in the literature and give
in-class expositions.