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\noindent This work is concerned with the extent to which certain
topological and algebraic phenomena, such as the mod one density
of the square roots of positive integers, or the existence of
irrationals and transcendentals, generalize from the standard
setting of $\langle \mathbb{R}, \mathbb{Z}\rangle$ to structures
of the form $\langle F,I\rangle$, where $F$ is an ordered field
and $I$ is an integer part (IP) for $F$. Here are some of the
highlights of the paper:
\noindent (i) Given an ordered field $F$, either all or none of
the IP's for $F$ (if any) are models of Open Induction, depending
on whether $F$ is dense in its real closure or not.
\noindent (ii) Density mod one of a subset is sensitive to a
certain relaxation of the notion of an IP in non-Archimedean real
closed fields with countable IP's. The weakened notion is that of
(additive subgroup) {\it integer sets} in the sense of J. Schmerl.
\noindent (iii) There is a real closed field $F$ with two IP's
$I_1$ and $I_2$ such that for some $\alpha\in I_2$ which is an
algebraic irrational with respect to $I_1$, the set
$\{u\alpha-\lfloor u\alpha \rfloor_{I_1}|u\in I_1\}$ is dense in
$[0,1)_F$.
\noindent (iv) All real closed fields $F$ of infinite
transcendence degree have proper dense real closed subfields over
which $F$ can be of any positive transcendence degree, up to and
including that of the transcendence degree of $F$ (they are real
closures of suitable proper dense subfields also). Together with a
result of van den Dries on the existence of real closed fields $F$
of infinite transcendence degree which possess certain IP's $I$
with the peculiar property that every element of $F$ is rational
with respect to $I$, this demonstrates the IP-sensitivity of
existence of transcendentals in $F$.
\noindent (v) Certain classes of real closed fields of infinite
transcendence degree such as Puiseux series, and all real closed
fields of finite transcendence degree, have algebraic irrationals
relative to every IP. Also, Cauchy complete real closed fields
have transcendentals relative to any IP.
\noindent (vi) For all $n\in\mathbb{N}^{\geq 1}$, there are real
closed fields $F$ and $K$ of transcendence degree $n$ such that
$F$ is algebraic over all its proper dense subfields, but $K$ is
algebraic over a proper dense subfield of itself, but
transcendental over another proper dense subfield of itself. This
and the previous two issues are also dealt with in the second
paper of F.-V. Kuhlmann in the present volume.
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