A connection between Inverse Problems and Nonstandard Analysis

Nonstandard analysis was born in the decade of 1960 an attempt to give a formal context to the Leibniz approach to differential calculus, in particular in order to provide a rigorous foundation to the notions of infinitesimal and infinite quantities in analysis. The theory soon found further nontrivial applications in several areas of mathematics including Analysis, Ergodic Theory, Geometric Group Theory, Probability Theory, Number Theory or Combinatorics. In this context, we find a remarkable proof by Renling Jin in 2007 of an extension of a result from Additive Number Theory, the so-called Freiman's Little Theorem, proving a longstanding conjecture for which the only known proof uses tools of nonstandard analysis. The objective of this master thesis is to give an account of this highly nontrivial proof by providing the necessary background on nonstandard analysis, with the aim of making the proof accessible to a wider readership outside the nonstandard analysis community