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Let D ⊆ R be closed and discrete and f : D n → R be such that f (D n) is somewhere dense. We show that (R, +, ·, f) defines Z. As an application, we get that for every α, β ∈ R with log α (β) / ∈ Q, the real field expanded by the two cyclic multiplicative subgroups generated by α and β defines Z.
We consider the expansion of the real field by a subgroup of a one-dimensional definable group satisfying a certain diophantine condition. The main example is the group of rational points of an elliptic curve over a number field. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that… (More)
We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property.
We present an elementary three pass algorithm for computing addition in Ostrowski numerations systems. When a is quadratic, addition in the Ostrowski numera-tion system based on a is recognizable by a finite automaton. We deduce that a subset of X ⊆ N n is definable in (N, +,V a), where V a is the function that maps a natural number x to the smallest… (More)
We give a criterion when an expansion of the ordered set of real numbers defines the image of (R, +, ·, N) under a semialgebraic injection. In particular, we show that for a non-quadratic irrational number α, the expansion of the ordered Q(α)-vector space of real numbers by N defines multiplication on R.
We study sets and groups definable in tame expansions of o-minimal structures. Let M = ⟨M, P ⟩ be an expansion of an o-minimal L-structure M by a dense set P. We impose three tameness conditions on M and prove a cone decomposition theorem for definable sets and functions in the realm of the o-minimal semi-bounded structures. The proof involves induction on… (More)
Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.
For first-order expansions of the field of real numbers, nondefinability of the set of natural numbers is equivalent to equality of topological and Assouad dimension on images of closed definable sets under definable continuous maps.
The theory of (R, <, +, Z, Za) is decidable if a is quadratic. If a is the golden ratio, (R, <, +, Z, Za) defines multiplication by a. The results are established by using the Ostrowski numeration system based on the continued fraction expansion of a to define the above structures in monadic second order logic of one successor. The converse that (R, <, +,… (More)