Expansions of o-minimal structures by fast sequences

Abstract

Let ℜ be an o-minimal expansion of (ℝ, <+) and (φ_k)_k∈ℕ be a sequence of positive real numbers such that lim_k→+∞f(φ_k)/φ_k+1=0 for every f:ℝ→ ℝ definable in ℜ. (Such sequences always exist under some reasonable extra assumptions on ℜ, in particular, if ℜ is exponentially bounded or if the language is countable.) Then (ℜ, (S)) is d-minimal, where S ranges over all subsets of cartesian powers of the range of φ.