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Existence of finite metric spaces with small-scale magnitude limit below 1 or infinite

Determine whether there exists a finite metric space X such that lim_{t→0} |tX| < 1 or lim_{t→0} |tX| = +∞, resolving the small-scale limit behavior of magnitude for finite metric spaces beyond the currently known case lim_{t→0} |tX| = α ≥ 1.

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Background

Prior work showed that for every α ≥ 1 there exists a finite metric space X with small-scale magnitude limit lim_{t→0} |tX| = α. The natural next question is whether limits strictly below 1 or diverging to +∞ can occur for finite spaces themselves (as opposed to along general paths).

In this paper, the authors construct paths in FMet converging to the one-point space whose limiting magnitudes achieve any extended real value except 1, but they explicitly note that this does not settle the existence of a finite space with lim_{t→0} |tX| < 1 or +∞.

References

In Thm.~3.8, it was shown that for every real number \alpha \geq 1, there exists a finite metric space X such that \lim_{t \to 0} |tX| = \alpha, but the question was left open of whether there exists X such that \lim_{t \to 0} |tX| < 1 or \lim_{t \to 0} |tX| = +\infty. That question is not answered here, but in \Cref{ex:perturb} we show that if one considers arbitrary paths (X_s){0 < s \ll 1} in FMet converging to the one-point space, then \lim{s \to 0} |X_s| can take any value in the extended real line.

— Is magnitude 'generically continuous' for finite metric spaces? (2501.08745 - Katsumasa et al., 15 Jan 2025) in Section 2, after Theorem 2.1 and before Example 2.2