  The continuous evolution of a wide variety of systems, including
continuous-time Markov chains and linear hybrid automata, can be described in
terms of linear differential equations. In this paper we study the decision
problem of whether the solution $\boldsymbol{x}(t)$ of a system of linear
differential equations $d\boldsymbol{x}/dt=A\boldsymbol{x}$ reaches a target
halfspace infinitely often. This recurrent reachability problem can
equivalently be formulated as the following Infinite Zeros Problem: does a
real-valued function $f:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ satisfying a
given linear differential equation have infinitely many zeros? Our main
decidability result is that if the differential equation has order at most $7$,
then the Infinite Zeros Problem is decidable. On the other hand, we show that a
decision procedure for the Infinite Zeros Problem at order $9$ (and above)
would entail a major breakthrough in Diophantine Approximation, specifically an
algorithm for computing the Lagrange constants of arbitrary real algebraic
numbers to arbitrary precision.
