  In the topological dynamical system $(X,T)$, a point $x$ simultaneously
approximates a point $y$ if there exists a sequence $n_1$, $n_2$, ... of
natural numbers for which $T^{n_i} x$, $T^{2n_i}x$, ..., $T^{k n_i} x$ all tend
to $y$. In 1978, Furstenberg and Weiss showed that every system possesses a
point which simultaneously approximates itself (a multiply recurrent point) and
deduced refinements of van der Waerden's theorem on arithmetic progressions. In
this paper, we study the denseness of the set of points that are simultaneously
approximated by a given point. We show that in a minimal nilsystem, all points
simultaneously approximate a $\delta$-dense set of points under a necessarily
restricted set of powers of $T$. We tie this theorem to the multiplicative
combinatorial properties of return-time sets, showing that all nil-Bohr sets
and typical return-time sets in a minimal system are multiplicatively thick in
a coset of a multiplicative subsemigroup of the natural numbers. This yields an
inhomogeneous multiple recurrence result that generalizes Furstenberg and
Weiss' theorem and leads to new enhancements of van der Waerden's theorem. This
work relies crucially on continuity in the prolongation relation (the closure
of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the
theory of rational points and polynomials on nilmanifolds developed by Leibman,
Green, Tao; and the machinery of topological characteristic factors developed
recently by Glasner, Huang, Shao, Weiss, and Ye.
