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.

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?

I am frequently amazed to discover that songs which I had believed to have been original are actually covers. It turns out, for instance, that TMBG’s “Istanbul (not Constantinople)” is a cover of a song from the 1950s.

Ironically, one might argue that Istanbul is itself a cover of Constantinople–and that argument (unifying form and content) reminds me of the language games played by Salt: Grain of Life, a book asserting that its very structure resembles the culinary crystal it purports to discuss.