For $X$ a metric space, and $S \subset X$, define the length spectrum of S to be $D_S := { d(x,y) : x, y \in S }$. It might be better to call this the “distance spectrum” or “distance set.”

Ian, during his Pizza seminar, gave the following definition: a set $S \subset \R^n$ is a $k$-distance set if $D_S$ has cardinality no greater than $k$. In words, the distances between points in a $k$-distance set take on no more than $k$ possible values.

The question that Ian answered is the following: how big can a $k$-distance set in $\R^n$ be? Clever linear algebra shows that the size grows polynomially in $n$ with degree $k$. A related exercise is the following: suppose $S \subset \R^n$ and $D_S$ is countable; prove that $S$ is countable.

Now here is my question: suppose $S \subset \R^n$ and $D_S$ is measurable with measure $m$. Can one then bound the measure of $S$? Ian asked this for the counting measure, but presumably one can get results for Lebesgue measure. Likewise, one can ask this for spaces other than $\R^n$.

All this talk of spectra has gotten me thinking very vaguely about a bunch of stuff–some random ideas! One context in which I have seen spectra is for lattices in Lie groups; I don’t know, but definitely ought to know how much control the spectrum exerts on the lattice. As a baby example, it is true that one can recover a lattice $\Lambda \subset \R^2$ from its length spectrum? Similarly, a Riemannian manifold has a length spectrum, and the “marked length rigidity conjecture” asks how much of the Riemannian structure is related to this. For information:

Furman, Alex. Coarse-geometric perspective on negatively curved manifolds and groups. 2002. 149–166. MR.

Finally, it is possible to define a “spectral distance” (I’m mis-using so many word here!) between two lattices in a Lie group. Namely, given $\Lambda_1, \Lambda_2 \subset G$, define $d(\Lambda_1, \Lambda_2) = d_{GH}(D_{\Lambda_1}, D_{\Lambda_2})$, i.e., the Hausdorff distance between their spectra. Though you’d probably want something slightly more refined (to count multiplicities). You could likewise say that two manifolds are “nearly isospectral” if their spectra are not so far apart in Gromov-Hausdorff distance. I have no idea whether this is a good idea; it probably isn’t. In any case,

Sunada, Toshikazu. Riemannian coverings and isospectral manifolds. Ann. of Math. (2) 1985. 169–186. MR.

constructs isospectral manifolds, and it would be interesting to know how easy it is to construct nearly isospectral manifolds. A braver person than I might conjecture that two manifolds are isospectral if they are $\epsilon$-nearly isospectral for small enough $\epsilon$.

At last, can one detect arithmeticity of a lattice from its spectrum? I suppose if I were really hip, I would ask: can a geometer hear arithmeticity? I think Sunada’s examples are all arithmetic?

Most mammals produce their own vitamin C, but humans carry a mutated form of the gene responsible for one of four enzymes enzymes necessary for vitamin C production, and so we humans must find it in our diets. In effect, every human being has a metabolic deficiency!

And in light of this wonderful news, why not ingest tremendously huge amounts of vitamin C?

In fact, I’d like to make this into a double-blind study of myself. Here is what I would like to do: randomly take either a placebo pill or a vitamin C pill (without my knowing which I took), and record the type of pill I took. At the end of the day, I would further record how I feel (as a number from 1 to 100, perhaps), and then do a regression to see if the type of pill I am taking is correlated with how I feel.

In fact, I should do this with all sorts of things in my life. Certainly I should be doing this with my caffeine intake, because I feel so convinced that I am much happier while drinking coffee, but that may only be an effect of the coffee–which is, wonderfully and exactly, the point.

The question is: how little can I spend to feed myself for one week? I ought to eat 2000 calories/day, so I’ll need to purchase 14,000 calories/week.

Here’s a “healthy” option: just eat apples. One ounce of apple has 15 calories, so I’ll need to eat 58 pounds of apples per week; I might be able to get this many apples for 29 dollars.

But I can do better! One “Take 5” candy bar is delicious and contains 210 calories, so I’ll need to eat 66 candy bars per week; in the best of all possible worlds, I might be able to purchase this many candy bars for 22 dollars.

I could buy a pound of mayonnaise for two dollars. Apparently a pound of mayonnaise has 3200 calories, so I could get more than enough calories for just ten dollars a week.

Presumably I could do significantly better with potatoes or with rice?

Tasha the Cat received a new toy–a plastic circle containing corrugated cardboard, with a ball stuck in a track. Watch her pounce!

On a recent plane trip, I was reading a very abridged version of (the ten thousand page long!) Church Dogmatics by Karl Barth, and I found something totally beautiful.

Believing God to be entirely “transcendent in contrast to all immanence” and “divine in contrast to everything human,” and reading (e.g., in Philippians 2:7) that Jesus is God having emptied himself, having made himself nothing, I concluded that God somehow hid his divinity in order that he might become human and, in that form, redeem humanity.

This is wrong. Karl Barth writes:

As God was in Christ, far from being against Him, or at disunity with Himself, He has put into effect the freedom of His divine love… He has therefore done and revealed that which corresponds to His divine nature…

His particular, and highly particularised, presence in grace, in which the eternal Word descended to the lowest parts of the earth and tabernacled in the man Jesus, dwelling in this one man in the fulness of His Godhead, is itself the demonstration and exercise of His omnipresence… His omnipotence is that of a divine plenitude of power in the fact that (as opposed to any abstract omnipotence) it can assume the form of weakness and impotence and do so as omnipotence, triumphing in this form…

From this we learn that the forma Dei [Philippians 2:6] consists in the grace in which God Himself assumes and makes His own the forma servi [Philippians 2:7].

–Church Dogmatics, Volume IV, Part 1, page 185 and following.

My cutting hardly does justice to the original text, so I’ll paraphrase.

Jesus shows that God is everywhere, because God is fully in him; this doesn’t undermine omnipresence, instead, it strengthens it: the abstract “God is everywhere” is emphasized by a particular “And look, God is there–it’s Jesus.” Similarly, Jesus shows that God is all-powerful, because God triumphed in him in spite of weakness.

I had been thinking that Jesus was God with a veil over his divinity, when in fact, Jesus is God proving just how totally divine he is. For a God who is Love, the incarnation isn’t a denial of himself, but an affirmation of who he had been all along. It is often said that Jesus proved his divinity by rising from the dead; it ought to be remembered that he proved his divinity by being able to be obedient to death in the first place.

This is a beautiful perspective from which to understand the hypostatic union; the monophysites believed that Jesus’ humanity undermined his divinity, while as Barth explains, Jesus’ two natures are not only compatible, but necessary. This is another example of the sort of paradoxical argument I usually find unreasonably compelling (e.g., Chesterton’s Orthodoxy or Kierkegaard (fear and trembling appears in Philippians 2:12–a coincidence?) or Hume’s compatibilist explanation of free will).

Like most things viewed with hindsight, this perspective isn’t radical, but I (and probably a lot of people) view the divine and human natures of Christ as, essentially, in conflict when, ironically, Jesus came to reconcile those two natures, and did so first in himself.