Applications of nonstandard analysis in additive number theory

This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density. 1 Prologue In this paper we use N for the set of all natural numbers. For any two integers a and b, [a; b] denotes the interval of all integers x between a and b (a 6 x 6 b). If S is an internal set, jSj denotes the internal cardinality ofS. The letters A; B; C will be used for sets of natural numbers and the letters i; h; k; m; n will be used for natural numbers. For a set A of natural numbers, the Shnirel'man density (A), the lower density d(A), the upper density d(A) and the Banach density BD(A) ofA are de ned as follows. (A) =inf n>1 jA \ [1;n]j n jA \ [1;n]j d(A) = lim inf n!