ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ - ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ
Λεωφ. Κνωσού, 714 09 Ηράκλειο. Τηλ: +30 2810393800, Fax +30 2810393881
Ο καθηγητής Ruzsa επισκέπτεται το Τμήμα Μαθηματικών στη «θέση Πηχωρίδη», τον από τα τέλη Σεπτεμβρίου έως τα μέσα Οκτωβρίου 2007. Θα δώσει σειρά ομιλιών με το ακόλουθο θέμα.
The idea to use Fourier series to study sets of integer is due to Hardy and Littlewood. They used it to approach classical topics like the Waring and Goldbach problems. In these cases the main difficulty is to estimate the resulting series, and the application of these estimates is fairly straightforward.
What can we do about a general set, say a finite, not too small subset of , or of , the set of residues modulo ? Here the only things we know immediately are the value at 0, which reflects the cardinality, and the suqare sum or integeral. Even this minimal knowledge can be used efficiently, and we demonstrate this on Bogolyubov's and Folner's theorems and Roth's result on 3-term arithmetical progressions.
In principle, the Fourier transform ``should know everything'', as it determines the original set uniquely via Fourier inversion. Still, it may be very difficult to extract this knowledge. An example is to find 4-term arithmetic progressions, where Gowers showed that the naive form of Roth's original approach is doomed to fail.
If we have a Fourier series (as a function), how do we see that it comes from a set and not from any function? It is easy to give a formal answer: it is equal to its convolution square. I know only one instance where this was used efficiently, and this is Bourgain's proof that a cosine sum assumes large negative values. I plan to tell a few words about this proof too.
The square mean does restrict the Fourier coefficients a lot, but there is more to say. We know that ther may be not too many large values. If there are not too few, they must be structured: they can be expressed as sums of very few among them. This is due to Bourgain and Chang.
The first talk will be an overview, the subsequent ones give some details.