Mathematics 540: Topics in Analytic Number Theory
Spring 2004
Instructor: I. Laba. Office: Math Bldg 239. Phone: 822 2450.
E-mail: ilaba@math.ubc.ca.
The course has been rescheduled to meet on Tuesdays and Thursdays 2 - 4,
except that we do not meet on those Thursdays on which the Number
Theory seminar is scheduled.
The general theme of this course will be "applications of Fourier
analysis in number theory". This will include topics in combinatorial and
additive number theory (Roth's theorem, Freiman's theorem) as well as
more classical analytic number theory (Hardy-Littlewood circle method,
Weyl's inequality). I am hoping to finish the course by presenting
Vinogradov's three-primes theorem - this is more technical and
complicated than the earlier topics, but it is also very interesting
and should make for a satisfying conclusion.
There will be no final exam or midterms. Instead, each student taking
this course for credit will be expected to present a short part of the
course material (1-2 hours). The presentation topics can be chosen from the
list below; if there is something else that you would like to present
here, please discuss it with me in advance.
There is a lot of active research going on in this field. Accordingly,
we will look at both classical results and some very recent work.
While the focus will be firmly on number theory, there will be various
bits of combinatorics and geometry mixed in. You may also expect
guest presentations from a few faculty and visitors working on closely
related topics.
Course outline (updated as of Feb. 22):
- Roth's theorem: a subset of Z of positive density must
contain a 3-term arithmetic progression.
Sources:
- W.T. Gowers: A new proof of Szemeredi's theorem, Geom. Funct.
Anal. 11 (2001), 465-588; also available from Tim Gowers's web page
(linked below)
Suggested additional topics:
- Roth's theorem for finite fields (ask Ben Green)
- Roth-type theorem for triangles in the plane (ask Jozsef Solymosi)
- Behrend's construction of a large set with no arithmetic progressions
- Freiman's theorem: a set of integers with a small sumset is
contained in a generalized arithmetic progression. In addition to
Fourier analysis, the proof also uses a fair bit of combinatorics
(Plunnecke-Ruzsa inequalities) and geometry of numbers (Minkowski's
theorem).
Sources:
- B.J. Green: Structure Theory of Set Addition,
ICMS Edinburgh Instructional Conference lecture notes, 2002; available
from Ben Green's web page (linked below)
- Nathanson's book, vol. II.
- Weyl's inequality: a classical method of estimating exponential
sums for uniformly distributed sequences.
Sources:
- Gowers-Verstraete lecture notes, see below.
- Montgomery's book.
- The Waring problem for squares: A set A of nonnegative integers
is called a basis of order k for N if every positive integer can
be written as a sum of k elements of A. Lagrange's theorem states that
every positive integer is a sum of four squares (i.e. squares are a basis
of order 4). Waring's problem is to extend this to cubes and higher powers
of integers. We will approach the problem via the "circle method", which
also provides an estimate on the number of representations of integers
as sums of squares.
Sources:
- Nathanson's book, vol. I.
- Vaughan's book.
- Schnirelman-Goldbach theorem: primes are a basis
of finite order. The proof includes an application of the
Selberg sieve.
Sources:
- Nathanson's book, vol. I.
- Vinogradov's three-primes theorem: every sufficiently large
odd integer is a sum of three primes. The proof incorporates many of the
techniques seen earlier in the course, and much more.
Sources:
- Gowers-Verstraete lecture notes.
- Davenport's book.
- Nathanson's book, vol. I.
- Szemeredi's theorem for 4-term arithmetic progressions,
if time allows. It is interesting to see why this is so much
more difficult than Roth's theorem, which we covered in the first
2 classes.
Sources:
- Gowers-Verstraete lecture notes.
This list of topics is not meant to be written in stone and may be modified
according to demand - but only if you make a good case. The sources
listed above are those that we are most likely to follow, but this
should not prevent you from consulting other books and papers, such as
those listed below and the references therein.
Lecture notes on the Web:
Books:
- H. Davenport, Multiplicative Number Theory, 2nd ed. (revised
by H. Montgomery), Springer-Verlag 1980.
- G.A. Freiman: Foundation of a structural theory of set addition, AMS, 1973
- H. Montgomery, Ten Lectures on the Interface between Analytic
Number Theory and Harmonic Analysis, AMS, 1994.
- M. Nathanson, Additive Number Theory, I: The Classical Bases;
II: Inverse Problems and the Geometry of Sumsets,
Springer-Verlag, 1996.
- R.C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge
Univ. Press, 1997 (note: the 2nd edition is substantially different
from the 1st).