An order-preserving Freiman 2-isomorphism is a map : X ! R such that (1) (a) < (b) if and only if a < b and (2) (a) + (b) = (c) + (d) if and only if a + b = c + d for any a, b, c, d 2 X. We show that for any A Z, if |A + A| ? K|A|, then there exists a subset A0 A such that the following holds: |A0 | K |A|, and there exists an order-preserving Freiman 2-isomorphism : A0 ! [c|A|, c|A|] Z where c depends only on K. Several applications are also presented.
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