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 . Given two sets and in , one says that *contains as a pattern* if there exists a similarity such that . (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 there is a positive-measure set 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 there exists a compact set of Hausdorff dimension which avoids 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 : for any three-point subset there exists a compact set of full Hausdorff dimension (that is, ) which does not contain 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 , let be its *divided difference set* (compare to the usual difference set ). Up to a similarity transformation, every nondegenerate triangle can be encoded by a complex number . Actually, is not unique: the numbers represent the same triangular pattern. According to the old book *Automorphic Forms* by Lester Ford, this incarnation of 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 as a fundamental domain for this group.

The author shows that if is compact and , then

is dense in . He asks whether the divided difference set of every compact 2-dimensional subset is dense in . Or maybe even is enough to make dense. Of course would not be enough since .

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.