Lecture Notes on Diophantine Analysis

The present book is a new, substantially enlarged, version of a previous set of lecture notes on diophantine analysis, published first in 2008, and then in revised form a few years later, by the *Edizioni della Scuola Normale*. The content mixes a number of rather classical results on diophantine equations and diophantine approximation, with the basic theory of heights and a few more recent results and applications of it. The exposition has been generally kept at an elementary and essentially self-contained level, focusing on some main ideas rather than finer technical results which can be obtained by similar methods. In fact, the book is addressed also to readers outside the relevant fields, with the hope that also more expert readers might find something relevant to them. The present second edition contains substantial new material, in the form of new sections, supplements, remarks, and exercises. The arguments for the exercises are developed in full by means of "hints", which in fact are much more than scattered suggestions, and practically contain complete details. Occasionally the remarks and the exercises contain miscellaneous results which have not been explicitly published elsewhere.