Is the space of reachable particle configurations dense?

TL;DR

This paper investigates the conditions under which a finite set of particles in Euclidean space can be moved arbitrarily close to any desired configuration using a specific jump move, establishing a density criterion based on the generated additive group.

Contribution

It characterizes when the space of reachable configurations is dense, linking it to the density of the additive group generated by initial position vectors.

Findings

Reachability is characterized by the density of the additive group generated by initial position vectors.

The space of configurations reachable via legal moves is dense if and only if the generated group is dense in .

Provides a necessary and sufficient condition for approximate configuration reachability in Euclidean space.

Abstract

Let $p_0,\ldots,p_n$ be a finite sequence of points in an Euclidean space $\R^d$. Suppose that there is a (pointlike) particle sitting at each point $p_i$. In a ``legal'' move, any one of them can jump over another, landing on the other side, at exactly the same distance. Under what circumstances can we guarantee that for any $\varepsilon>0$ and any other sequence of points $q_0,\ldots, q_n\in\R^d$, there is a finite sequence of legal moves that takes the particle at $p_i$ to the $\varepsilon$-neighborhood of $q_i$, simultaneously for every $i$? We prove that this is possible if and only if the additive group generated by the vectors $p_1-p_0,\ldots,p_n-p_0$ is dense in $\R^d$.