C3.10 Additive and Combinatorial Number Theory - Material for the year 2020-2021

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Dr Joni Teravainen
General Prerequisites: 

The only essential prerequisite course is Part A ASO: Number Theory. Attendance at C3.8 Analytic Number Theory will certainly be helpful, but is not essential. Fourier transforms will appear at several points in the course, but we will develop what we need from scratch.
However, the course B4.3 Distribution Theory and B4.4 Fourier Analysis may provide some useful context.

Course Term: 
Course Weight: 
1.00 unit(s)
Course Level: 

Assessment type:

Course Overview: 

The aim of this course is to present classic results in additive and combinatorial number theory, showing how tools from a variety of mathematical areas may be used to solve number-theoretical problems. Highlights will include the classical theorem of Lagrange that every number is the sum of four squares, results on Waring's problem (every number is the sum of $s$ perfect $k$th powers, where $s$ is bounded as a function of $k$) and Roth's theorem that sets of integers with positive upper density contain infinitely many 3-term arithmetic progressions, the first interesting case of Szemerédi's theorem. We will also discuss a celebrated theorem of Freiman describing the structure of finite sets of integers $A$ for which the number of distinct sums $\{a + a' : a, a' \in A\}$ is not much larger than the size of $A$. If time allows, we will hint at the application of this, in the work of Tim Gowers, to Szemerédi's theorem for progressions of length 4.

Course Synopsis: 

Sums of squares. Every prime congruent to 1 modulo 4 is a sum of two squares. Every natural number is the sum of four squares. *Discussion of sums of three squares*.

Waring's problem on sums of $k$th powers. The Hardy-Littlewood circle method, major and minor arcs. Hua's lemma. Estimates for Weyl sums. Asymptotic formula for the number of representations of $n$ as a sum of $s$ $k$th powers when $s$ is sufficiently large in terms of $k$. *Discussion of Vinogradov's theorem that every sufficiently large odd number is the sum of 3 primes*

Roth's theorem on arithmetic progressions of length 3.

Basic sumset estimates. Bohr sets and Bogolyubov's theorem. Geometry of numbers and Minkowski's second theorem. Freiman's theorem on sets with small doubling constant.
*Discussion of Gowers's work on Szemerédi's theorem for progressions of length 4 and longer*.

Reading List: 

Full printed notes will be produced for the course. The material in the first part of the course (and a lot more) is covered at a rapid pace in R. Vaughan, The Hardy-Littlewood Method.
Everything in the second part of the course (and considerably more) is covered in T. Tao and V. Vu, Additive Combinatorics.