Inverse problem for cuts (Q2458891)

A set \(X \subseteq \mathbb N\) is called a bi-arithmetic progression of ratio \(d\) if \(X=I_0 \cup I_1\) for two arithmetic progressions \(I_0\) and \(I_1\) with a common ratio \(d\) such that \(I_0+I_0\), \(I_0+I_1\), \(I_1+I_1\) are pairwise disjoint. A set \(X \subseteq \mathbb N\) is called a tri-arithmetic progression of ratio \(d\) if \(X=I_0 \cup I_1 \cup I_2\) for three arithmetic progressions \(I_0\), \(I_1\) and \(I_2\) with a common ratio \(d\) such that exactly five of the six sets \(I_i+I_j\) for \(i \leq j\) in \(\{0,1,2\}\) are pairwise disjoint. The author uses nonstandard analysis techniques to approach the following conjecture: There exists a natural number \(K\) such that for any finite set \(A\) of integers with cardinality \(| A | = k > K\) and \(| A+A | = 3k-3+b\) for \(0\leq b < {1\over3}k -2\), \(A\) is either a subset of an arithmetic progression of length at most \(2k-1+2b\) or a subset of a bi-arithmetic progression of length at most \(k+b\). The main result states that if \(U\) is a cut in \(^*\mathbb N\) with uncountable cofinality and \(A_0 \subseteq U\) is \(U\)-internal such that \(0 \in A\) and \(0 < \underline{d}_U(A_0) = \alpha < {3\over 5}\), then one of the following is true: (a) \(A_0\) is a subset of an arithmetic progression of ratio \(g > 1\). (b) \(A_0\) is a subset of a bi-arithmetic progression of the form \(F+gU\) where \(g >2\) and \(F = \{0, a\} \subseteq [0, g-1]\). (c) \(A_0\) is a subset of a tri-arithmetic progression of the form \(F+gU\) where \(g>4\) and \(F = \{0, a_1, a_2\} \subseteq [0, g-1]\). (d) There is \(\varepsilon > 0\) and there are sufficiently large \(x \in A_0\) such that \((A_0+A_0)(2x) > ({10\over 3} + \varepsilon)A_0(x)\).