“Full dimensional sets without given patterns” by Péter Maga

The footnote says that the paper is a part of the authors’ Master’s thesis at Eötvös Loránd University. Well, they write neat Master’s theses over in Budapest. The paper is available online as a preprint and in final version (by subscription). I rewrote my Zentralblatt review into something more resembling a blog post.

In analogy with additive number theory, I am going to define additive geometric measure theory as the search for arithmetic/geometric patterns in sufficiently large subsets of $\mathbb R^n$. Given two sets $A$ and $P$ in $\mathbb R^n$, one says that $A$ contains $P$ as a pattern if there exists a similarity $\phi\colon \mathbb R^n\to\mathbb R^n$ such that $\phi(P)\subset A$. (A similarity is a map that multiplies all distances by the same nonzero factor.)

It is not hard to see (via Lebesgue density theorem) that a set of positive measure contains every finite set as a pattern. Erdős conjectured that finiteness is essential here: for any infinite pattern $P$ there is a positive-measure set $A$ which avoids it. This conjecture of Erdős is still open. The paper investigates the opposite direction: the patterns are finite, but the sets which contain (or avoid them) are smaller. There is nothing interesting about 1- and 2-point patterns, so the first case of interest is a 3-point pattern.

Tamás Keleti proved that for every three-point subset $P\subset \mathbb R$ there exists a compact set $A\subset \mathbb R$ of Hausdorff dimension $1$ which avoids $P$ as a pattern (actually, his sets can avoid countably many such patterns at once). One of the results of the present article achieves the same goal in $\mathbb R^2$: for any three-point subset $P\subset \mathbb R^2$ there exists a compact set $A\subset \mathbb R^2$ of full Hausdorff dimension (that is, $2$) which does not contain $P$ as a pattern. The higher dimensional version remains open.

The two dimensional case is special because the triangle-avoiding property can be related to the arithmetics of complex numbers. Given a set $A\subset \mathbb C$, let $\mathcal T(A)=\left\{\frac{z-x}{y-x} \colon x,y,z\in A, x\ne y\right\}$ be its divided difference set (compare to the usual difference set $\mathcal D(A)=\left\{{y-x} \colon x,y\in A\right\}$). Up to a similarity transformation, every nondegenerate triangle can be encoded by a complex number $\zeta\ne 0$. Actually, $\zeta$ is not unique: the numbers $\zeta, \zeta^{-1}, 1-\zeta, (1-\zeta)^{-1}, 1-\zeta^{-1}, (1-\zeta^{-1})^{-1}$ represent the same triangular pattern. According to the old book Automorphic Forms by Lester Ford, this incarnation of $D_3$ is called the group of anharmonic ratios, or maybe just the anharmonic group. I don’t remember seeing the term in more recent literature, but this does not say much. Anyway, one can take the triangle $\{z\colon \mathrm{Im}\, z>0, |z|>1, |z-1|>1\}$ as a fundamental domain for this group.

The author shows that if $A\subset \mathbb R$ is compact and $\mathrm{dim} A=1$, then
$\mathcal T(A)$ is dense in $\mathbb R$. He asks whether the divided difference set of every compact 2-dimensional subset $A\subset \mathbb C$ is dense in $\mathbb C$. Or maybe even $\mathrm{dim}\, A>1$ is enough to make $\mathcal T(A)$ dense. Of course $\mathrm{dim}\, A = 1$ would not be enough since $\mathcal T(\mathbb R)=\mathbb R$.

Some of the results of this paper were later put into a larger framework by András Máthé in Sets of large dimension not containing polynomial configurations.

Gromov’s “Hilbert volume in metric spaces I”

Just finished writing a summary of Gromov’s Hilbert Volume in Metric Spaces, Part 1 for Zentralblatt. Not an easy task to summarize such an article, and I essentially limited myself to the introductory part. But at least I contributed a Parseval frame analogy, which is not explicit in the article. I like frames in general, and tight/Parseval frames most of all.

And since the Zentralblatt server for review submission is down at the moment, the outlet for this text defaults to my blog.

Let $Df$ denote the Jacobian matrix of a differentiable map $f\colon \mathbb R^n\to\mathbb R^n$. One way to quantify the infinitesimal dilation of $f$ is to consider the operator norm $\|Df\|$. This corresponds to the local form of the Lipschitz constant $\mathrm{Lip}\,f=\sup_{a,b} \frac{|f(a)-f(b)|}{|a-b|}$, and thus makes sense in general metric spaces. Another natural, and often more convenient way to measure dilation is the Hilbert-Schmidt norm $\|Df\|_{HS}$. However, the latter does not immediately generalize to metric spaces. The present article developes such a generalization and uses it to derive several known previously results in a unified and elegant way.

Instead of trying to describe the construction in full generality, let us consider the special case of Lipschitz maps $f\colon X\mapsto \mathbb R^n$, where $X$ is a metric space. Let $\mu$ be a measure on the set $\mathcal P$ of all rank-$1$ projections, which can be identified on the $(n-1)$-dimensional projective space. Of particular importance are the measures $\mu$ for which $\int_{\mathcal P}p(x)\,d\mu = x$ for all $x\in \mathbb R^n$. Such a measure is called an axial partition of unity; a related term in harmonic analysis is a Parseval frame. The $L_2$-dilation of $f$ with respect to $\mu$ is $\|\mathrm{dil}^* f\|_{L_2(\mu)}=\left(\int_{\mathcal P} \mathrm{Lip}^2(p\circ f) \,d\mu(p)\right)^{1/2}$. In terms of frames, this definition means that one applies the analysis operator to $f$ and measures the Lipschitz constant of the output. Another approach is to use the synthesis operator: consider all Lipschitz maps $\tilde f\colon X\to L_2(\mathcal P,\mu)$ from which $f$ can be synthesized and define $\|\widetilde{\mathrm{dil}}^* f\|_{L_2(\mu)}=\inf_{\tilde f} \left(\int_{\mathcal P} \mathrm{Lip}^2(\tilde f(\cdot,p))\,d\mu(p) \right)^{1/2}$.

For every axial partition of unity one has $\|\widetilde{\mathrm{dil}}^* f\|_{L_2(\mu)} \le \|\mathrm{dil}^* f\|_{L_2(\mu)}$ because the composition of analysis and synthesis recovers $f$. Taking the infimum over all axial partitions of unity $\mu$ yields minimal $L_2$-dilations $\|\min\mathrm{dil}^* f\|_{L_2}$ and $\|\min\widetilde{\mathrm{dil}}^* f\|_{L_2}$.

For linear maps between Euclidean spaces the minimal $L_2$-dilation of either kind is exactly the Hilbert-Schmidt norm. For non-linear maps they need to be localized first, by taking restrictions to small neighborhoods of a point. The concept turns out to be useful, e.g., for proving volume comparison theorems. The author proves an elegant form of F. John’s ellipsoid theorem in terms of $\|\min\mathrm{dil}^* f\|_{L_2}$, recasts the Burago-Ivanov proof of the Hopf conjecture [Geom. Funct. Anal. 4, No.3, 259-269 (1994; Zbl 0808.53038)] in these new terms, and presents further extensions and applications of his approach.