Submission date: 29 January 2017

Abstract:

We determine the sets definable in expansions of the ordered real additive group by generalized Cantor sets. Given a natural number r ≥ 3, we say a set C is a generalized Cantor set in base r if there is a non-empty K ⊆ {1,...,r-2} such that C is the set of those numbers in [0,1] that admit a base r expansion omitting the digits in K.
While it is known that the theory of an expansion of the ordered real additive group by a single generalized Cantor set is decidable, we establish that the theory of an expansion by two generalized Cantor sets in multiplicatively independent bases is undecidable.

Mathematics Subject Classification: Primary 03B25 Secondary 03B70, 03C64, 28A80

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Full text arXiv 1701.08426: pdf, ps.