“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.

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