Supervisors info:
Απόστολος Γιαννόπουλος, Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών (επιβλέπων)
Δημήτριος Γατζούρας, Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Κωνσταντίνος Τύρος, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Summary:
The present thesis consists of two parts. In the first part of the thesis we present three different solutions of Waring's problem. At first we take a look at Hilbert's proof. Next, we present a proof of the problem by using the Hardy-Littlewood circle method and analytic tools like Weyl's inequality. Lastly, we study a combinatorial proof by Linnik that uses the concept of Schnirelmann's density which we present.
In the second part we present a later proof of Freiman's theorem by Rusza. We start with some basic estimations of set sums. Consequently we present tools from graph theory (Plunnecke's theorem) and geometry of numbers (Minkowski's theorems). Finally we sumbit the proof of the main theorem.
