ETD

Archivio digitale delle tesi discusse presso l'Università di Pisa

Tesi etd-06272017-175221


Tipo di tesi
Tesi di laurea magistrale
Autore
MARANGIO, LUIGI
URN
etd-06272017-175221
Titolo
Arithmetic Progressions in a Set of Positive Density
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Di Nasso, Mauro
Parole chiave
  • Additive Combinatorics
  • Szemeredi's Theorem
  • Ergodic Theory
  • Ramsey Theory
Data inizio appello
14/07/2017
Consultabilità
Completa
Riassunto
An arithmetic progression is a sequence of numbers such that the difference between the
consecutive terms is constant; how many arithmetic progressions there are in a subset of the natural numbers?

In 1927 van der Waerden proved a celebrated theorem, which states that if the natural numbers are partitioned into finitely many classes (the ``colors") then at least one of these classes contains arbitrarily long arithmetic progressions.
In the same period of van der Waerden, Ramsey proved that in any infinite complete k-uniform hypergraph colored by a finite numbers of colors, there is a monochromatic infinite complete k-uniform section hypergraph. The van der Waerden and the Ramsey theorems, as they are now called, initiated
the field of Ramsey theory, that is (roughly speaking) the study of different
notions of largeness for parts of the natural numbers (or more general structure such as semigroup) and how these notions are linked to each other.
A strengthening of a completely different kind was conjectured by Erdos and Turán in 1936. They realised that it ought to be possible to find arithmetic progressions of length k in any sufficiently dense set of integers.
In 1953, Roth solved the conjecture of Erdos and Turán in the case k=3; Roth gave a beautiful argument using exponential-sum estimates, but his approach seemed not to generalize.

The case k=4 was solved only in 1969 by Szemerédi with a
complex combinatoric argument. Finally the conjecture was
proved by Szemerédi in 1974; Szemerédi’s theorem is one of the milestones of combinatorics.

A second proof of Szemerédi’s theorem was given by Furstenberg in 1977, by reducing it, thanks to the so called Furstenberg correspondence principle, to a problem about measure preserving systems.
In this thesis we will discuss the approach of Roth and Furstenberg and we will show that Roth’s argument can be generalized, as Gowers pointed out in 1998 for the case k=4, and for the general case in 2001.
File