For any infinite abelian group G and any integers m_1,...,m_H, finite sets A,B exist such that |hA| - |hB| = m_h for every h.
Walker, A tight structure theorem for sumsets
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Finer control on relative sizes of iterated sumsets
For any infinite abelian group G and any integers m_1,...,m_H, finite sets A,B exist such that |hA| - |hB| = m_h for every h.