# “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.