Sumsets in sets of positive density

A central theme in the mathematical field of arithmetic combinatorics is the extent to which size alone can guarantee additive structure. A typical question along these lines is: for which arithmetic patterns is it the case that any large enough collection of numbers is guaranteed to contain a copy of the pattern? The goal of the proposed research is to push towards a new and exciting result in the field: that every set of natural numbers that constitutes a positive proportion of all natural numbers contains the sum of any finite number of infinite sets.