  The Semialgebraic Orbit Problem is a fundamental reachability question that
arises in the analysis of discrete-time linear dynamical systems such as
automata, Markov chains, recurrence sequences, and linear while loops. An
instance of the problem comprises a dimension $d\in\mathbb{N}$, a square matrix
$A\in\mathbb{Q}^{d\times d}$, and semialgebraic source and target sets
$S,T\subseteq \mathbb{R}^d$. The question is whether there exists $x\in S$ and
$n\in\mathbb{N}$ such that $A^nx \in T$. The main result of this paper is that
the Semialgebraic Orbit Problem is decidable for dimension $d\leq 3$. Our
decision procedure relies on separation bounds for algebraic numbers as well as
a classical result of transcendental number theory---Baker's theorem on linear
forms in logarithms of algebraic numbers. We moreover argue that our main
result represents a natural limit to what can be decided (with respect to
reachability) about the orbit of a single matrix. On the one hand,
semialgebraic sets are arguably the largest general class of subsets of
$\mathbb{R}^d$ for which membership is decidable. On the other hand, previous
work has shown that in dimension $d=4$, giving a decision procedure for the
special case of the Orbit Problem with singleton source set $S$ and polytope
target set $T$ would entail major breakthroughs in Diophantine approximation.
