Very briefly, the question we consider is the following. Let be a closed set of Hausdorff dimension . Given a system of matrices for some $m \geq n$, must contain a “non-trivial” k-point configuration
for some and ?
In general, the answer is no, even when . For instance, Keleti has constructed 1-dimensional subsets of that do not contain a similar copy of any given triple of points (in fact, his construction can avoid all similar copies of all such triples from a given sequence), as well as 1-dimensional subsets of that do not contain any non-trivial “parallelograms” . In , given any three distinct points , Maga has constructed examples of sets of dimension 2 that do not contain any similar copy of the triangle ; he also constructed sets of full dimension in , for any , that do not contain non-trivial parallelograms.
Additive combinatorics suggests, however, that sets that are “random” in an appropriate sense should he better behaved in that regard. Along these lines, Malabika Pramanik and I proved in an earlier paper that if has dimension close enough to 1, and if it also supports a measure obeying appropriate dimensionality and Fourier decay estimates, then must contain a non-trivial 3-term arithmetic progression. The same proof applies to any other configuration , with the dimension bound depending on the choice of configuration.
This paper gives a multidimensional analogue of that result. We define, via conditions on the matrices , a class of configurations that can be controlled by Fourier-analytic estimates. (Roughly, they must have enough degrees of freedom, and they must be “non-degenerate” in an appropriate sense.) For such , if has dimension close enough to , and if it supports a measure with dimensionality and Fourier decay conditions similar to those in my paper with Pramanik, then must indeed contain a non-trivial configuration as in (1).
The main new difficulty is dealing with the complicated geometry of the problem. There’s a lot of linear algebra, multiple coordinate systems, multilinear forms, and a lot of estimates on integrals where the integrand decays at different rates in different directions. At one point, we were actually using a partition of unity similar to those I remembered from my work in multiparticle scattering theory a very long time ago. That didn’t make it into the final version, though – we found a better way.
I won’t try to state the conditions on here – they’re somewhat complicated and you’ll have to download the paper for that – but I’ll mention a few special cases of our theorem.
- Triangles in . Let be three distinct points in the plane. Suppose that satisfies the assumptions of our main theorem. Then must contain three distinct points such that the triangle is a similar (possibly rotated) copy of the triangle .
- Colinear triples in . Let be three distinct colinear points in . Assume that satisfies the assumptions of our main theorem. Then must contain three distinct points that form a similar image of the triple .
- Parallelograms in . Assume that satisfies the assumptions of our main theorem. Then contains a parallelogram , where the four points are all distinct.