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.