Conant's Metric Spectra Problem

In this paper, we try to minimize the scope of possible unique metric spectra up to equivalence. While it is well known that every spectra $S\subseteq {\mathbb{R}}^{+}$ is equivalent to a spectra $T\subseteq \mathbb{N}$, it has remained open if $T$ could also maintain a desirable combinatorial form. Conant questioned if $T=\{t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_n{\}}_{<}$ could be taken such that $2^i-1\le t_i\le 2^n-1$. In this paper, we come to two partial answers. The first is that the largest element $t_n$ can be chosen such that $t_n\le 2^n$. Approximating a full solution, we also observe $T$ with the combinatorial form $2^i\le t_i\le 2^{n+1}$. Our methods are rather unique in the field as we utilize linear optimization and polygonal geometry to achieve our results. Our work aims to approach a full characterization of metric spectra, and simplify future computational endeavors in the field.