For any k-automatic non-eventually-periodic set X, the structure (N, +, X) either defines all k-automatic sets or has exactly the same definable sets as (N, +, k^N).
¨Uber die vollst¨ andigkeit eines gewissen systems der arithmetik ganzer zahlen, in welchem die addition als einzige operation hervortritt.C
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A Dichotomy for $k$-automatic expansions of Presburger Arithmetic
For any k-automatic non-eventually-periodic set X, the structure (N, +, X) either defines all k-automatic sets or has exactly the same definable sets as (N, +, k^N).