The n-Category Café

Asymptotic results

As you may recall, the magnitude $\mathrm{mag}(X)$ of a finite subset $X\subseteq \mathbb{R}^n$ is easy to define: let $w:X\to \mathbb{R}$ be a function which satisfies

$$\sum_{y\in X} \mathrm{e}^{-\mathrm{d}(x,y)} w(y) = 1 \quad \text{for all}\ x \in X.$$

Such a function is called a weighting. Then the magnitude is defined as the sum of the weights:

$$\mathrm{mag}(X) = \sum_{x \in X} w(x).$$

For a compact subset $X$ of $\mathbb{R}^n$, Mark Meckes shows that all reasonable definitions of magnitude are equal to what you get by taking the supremum of the magnitudes of all finite subsets of $X$:

$$\mathrm{mag}(X) = \sup \{\mathrm{mag}(\Xi) : \Xi \subset X \ \text{finite} \} .$$

Unfortunately, few explicit computations of the magnitude for a compact subset of $\mathbb{R}^n$ are known.

Instead of the magnitude of an individual set $X$, it proves fruitful to study the magnitude of dilates $R\cdot X$ of $X$, for $R\gt 0$. Here the dilate $R\cdot X$ means the space $X$ with the metric scaled by a factor of $R$. We can vary $R$ and this gives rise to the magnitude function of $X$:

$$\mathcal{M}_X\colon (0,\infty)\to \mathbb{R};\quad\mathcal{M}_X(R) := \mathrm{mag}(R\cdot X)\quad\text{for }\ R \gt 0.$$

Tom and Simon conjectured a relation between the magnitude function of $X$ and its intrinsic volumes $(V_i(X))_{i=0}^n$. The intrinsic volumes of subsets of $\mathbb{R}^n$ generalize notions such as volume, surface area and Euler characteristic, with $V_n(X)=\text{vol}_n(X)$ and $V_0(X)=\chi(X)$.

Convex Magnitude Conjecture. Suppose $X \subseteq \mathbb{R}^n$ is compact and convex, then the magnitude function is a polynomial whose coefficients involve the intrinsic volumes:
$$\mathcal{M}_X(R) = \sum_{i=0}^n \frac{V_i(X)}{i! \,\omega_i} R^n,$$ where $V_i(X)$ is the $i$-th intrinsic volume of $X$ and $\omega_i$ the volume of the $i$-dimensional ball.

The conjecture was disproved by Barceló and Carbery (see also this post). They computed the magnitude function of the $5$-dimensional ball $B_5$ to be the rational function $$\mathcal{M}_{B_5}(R)=\frac{R^5}{5!} +\frac{3R^5+27R^4+105R^3+216R+72}{24(R+3)}.$$ In particular, the magnitude function is not even a polynomial for $B_5$. Also, the coefficient of $R^4$ in the asymptotic expansion of $\mathcal{M}_{B_5}(R)$ as $R \to \infty$ does not agree with the conjecture.

Nevertheless, for any smooth, compact domain in odd-dimensional Euclidean space, $X\subseteq\mathbb{R}^n$ (in other words, the closure of an open bounded subset with smooth boundary), for $n=2m-1$, our main result shows that a modified form of the conjecture holds asymptotically as $R \to \infty$.

Theorem A. Suppose $n=2m-1$ and that $X\subseteq \mathbb{R}^n$ is a smooth domain.

1. There exists an asymptotic expansion of the magnitude function: $$\mathcal{M}_X(R) \sim \frac{1}{n!\omega_n}\sum_{j=0}^\infty c_j(X) R^{n-j} \quad \text{as }\ R\to \infty$$ with coefficients $c_j(X)\in\mathbb{R}$ for $j=0,1,2,\ldots$.

2. The first three coefficients are given by $$\begin{aligned} c_0(X)&=\text{vol}_n(X),\\ c_1(X)&=m\text{vol}_{n-1}(\partial X),\\ c_2(X)&=\frac{m^2}{2} (n-1)\int_{\partial X} H \ \mathrm{d}S, \end{aligned}$$ where $H$ is the mean curvature of $\partial X$. (The coefficient $c_0$ was computed by Barceló and Carbery.) If $X$ is convex, these coefficients are proportional to the intrinsic volumes $V_{n}(X)$, $V_{n-1}(X)$ and $V_{n-2}(X)$ respectively.

3. For $j\geq 1$, the coefficient $c_j(X)$ is determined by the second fundamental form $L$ and covariant derivative $\nabla_{\partial X}$ of $\partial X$: $c_j(X)$ is an integral over $\partial X$ of a universal polynomial in the entries of $\nabla_{\partial X}^k L$, $0 \leq k\leq j-2$. The total number of covariant derivatives appearing in each term of the polynomial is $j-2$.

4. The previous part implies an asymptotic inclusion-exclusion principle: if $A$, $B$ and $A \cap B \subset \mathbb{R}^n$ are smooth, compact domains, $$\mathcal{M}_{A \cup B}(R) - \mathcal{M}_A(R) - \mathcal{M}_B(R) + \mathcal{M}_{A \cap B}(R) \to 0 \quad \text{as }\ R \to \infty$$ faster than $R^{-N}$ for all $N$.

If you’re not familiar with the second fundamental form, you should think of it as the container for curvature information of the boundary relative to the ambient Euclidean space. Since Euclidean space is flat, any curvature invariant of the boundary (satisfying reasonable symmetry conditions) will only depend on the second fundamental form and its derivatives.

Note that part 4 of the theorem does not imply that an asymptotic inclusion-exclusion principle holds for all $A$ and $B$, even if $A$ and $B$ are smooth, since the intersection $A \cap B$ usually is not smooth. In fact, it seems unlikely that an asymptotic inclusion-exclusion principle holds for general $A$ and $B$ without imposing curvature conditions, for example by means of assuming convexity of $A$ and $B$.

The key idea of the short proof relates the computation of the magnitude to classical techniques from geometric and semiclassical analysis, applied to a reformulated problem already studied by Meckes and by Barceló and Carbery. Meckes proved that the magnitude can be computed from the solution to a partial differential equation in the exterior domain $\mathbb{R}^n\setminus X$ with prescribed values in $X$. A careful analysis by Barceló and Carbery refined Meckes’ results and expressed the magnitude by means of the solution to a boundary value problem. We refer to this boundary value problem as the “Barceló-Carbery boundary value problem” below.

Meromorphic extension of the magnitude function

Intriguingly, we find that the magnitude function $\mathcal{M}_X$ extends meromorphically to complex values of the scaling factor $R$. The meromorphic extension was noted by Tom for finite metric spaces and was observed in all previously known examples.

Theorem B. Suppose $n=2m-1$ and that $X\in \mathbb{R}^n$ is a smooth domain.

• The magnitude function $\mathcal{M}_X$ admits a meromorphic continuation to the complex plane.

• The poles of $\mathcal{M}_X$ are generalized scattering resonances, and each sector $\{z : |\arg(z)| \lt \theta \}$ with $\theta \lt \frac{\pi}{2}$ contains at most finitely many of them (all of them outside $\{z : |\arg(z)|\lt {\textstyle \frac{\pi}{n+1}}\}$).

The ordinary notion of scattering resonances comes from studying waves scattering at a compact obstacle $X\subseteq \mathbb{R}^n$. A scattering resonance is a pole of the meromorphic extension of the solution operator $(R^2-\Delta)^{-1}$ to the Helmholtz equation on $\mathbb{R}^n\setminus X$, with suitable boundary conditions. These resonances determine the long-time behaviour of solutions to the wave equation and are well studied in geometric analysis as well as throughout physics. The Barceló-Carbery boundary value problem is a higher order version of this problem and studies solutions to $(R^2-\Delta)^{m}u=0$ outside $X$. In dimension $n=1$ (i.e. $m=1$), the Barceló-Carbery problem coincides with the Helmholtz problem, and the poles of the magnitude function are indeed scattering resonances. As in scattering theory, one might hope to find detailed structure in the location of the poles. A plot of the poles and zeros in case of the $21$-dimensional ball is given at the top of the post.

The second part of this theorem is sharp. In fact, the poles do not need to lie in any half plane. Using the techniques of Barceló and Carbery we observe that the magnitude function of the 3-dimensional spherical shell $X=(2B_3)\setminus B_3^\circ$ is not rational and contains an infinite discrete sequence of poles which approaches the curve $\mathrm{Re}(R)= \log(|\mathrm{Im}(R)|)$ as $\mathrm{Re}(R) \to \infty$. Here’s a plot of the poles of $\mathcal{M}_{X}$ with $|\mathrm{Im}(R)|\lt 30$.

The magnitude function of $X$ is given by $$\mathcal{M}_X(R)=(7/6)R^3 +5R^2 + 2R + 2 + \frac{e^{-2R}(R^2+1)+2R^3-3R^2+2R-1}{\sinh 2R -2R}.$$

[EDIT: The above formula has been corrected, following comments below.]

Our techniques extend to compact domains with a $C^k$ boundary, as long as $k$ is large enough. In this case, the asymptotic inclusion-exclusion principle takes the form that $$\mathcal{M}_{A \cup B}(R) - \mathcal{M}_A(R) - \mathcal{M}_B(R) + \mathcal{M}_{A \cap B}(R) \to 0 \quad \text{as}\ R \to \infty$$ faster than $R^{-N}$ for an $N=N(k)$.