Logic and Analysis 1 (1):61-89 (2007)
|Abstract||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 Î± strictly between 0 and 3/5. We show that either there exists a standard real ε >Â 0 and there are sufficiently large x in A such that | (A+A) â© [0, 2x]| > (10/3+ ε $ ) | A â© [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|
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