Putnam 2023, Problem A-4

Let \( v_1, \ldots, v_{12} \) be unit vectors in \( \mathbb{R}^3 \) from the origin to the vertices of a regular icosahedron. Show that for every vector \( v \in \mathbb{R}^3 \) and every \( \varepsilon > 0 \), there exist integers \( a_1, \ldots, a_{12} \) such that \( \|a_1v_1 + \cdots + a_{12}v_{12} - v\| < \varepsilon \).

Solution:
The assumption that all vertices of the icosahedron correspond to vectors of the same length forces the center of the icosahedron to lie at the origin, since the icosahedron is inscribed in a unique sphere. Since scaling the icosahedron does not change whether or not the stated conclusion is true, we may choose coordinates so that the vertices are the cyclic permutations of the vectors $(\pm \frac{1}{2}, \pm \frac{1}{2}\phi, 0)$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. The subgroup of $\mathbb{R}^3$ generated by these vectors contains $G \times G \times G$ where $G$ is the subgroup of $\mathbb{R}$ generated by $1$ and $\phi$. Since $\phi$ is irrational, it generates a dense subgroup of $\mathbb{R}/\mathbb{Z}$; hence $G$ is dense in $\mathbb{R}$, and so $G \times G \times G$ is dense in $\mathbb{R}^3$, proving the claim.