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 $dinmathbb{N}$, a square matrix $Ainmathbb{Q}^{dtimes d}$, and semialgebraic source and target sets $S,Tsubseteq mathbb{R}^d$. The question is whether there exists $xin S$ and $ninmathbb{N}$ such that $A^nx in T$. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension $dleq 3$. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory---Bakers 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.
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