Logic and Analysis Inverse problem for cuts

Let U be an initial segment of ∗N closed under addition (such U is called a cut) with uncountable cofinality and A be a subset of U , which is the intersection of U and an internal subset of ∗N. Suppose A has lower U -density α strictly between 0 and 5 . We show that either there exists a standard real > 0 and there are sufficiently large x in A such that |(A+ A)∩[0, 2x]| > ( 10 3 + ) |A∩[0, x]| or A is a large subset of an arithmetic progression of difference greater than 1 or A is a large subset of the union of two arithmetic progressions with the same difference greater than 2 or A is a large subset of the union of three arithmetic progressions with the same difference greater than 4.