“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.