One more paper finished: “Finite configurations in sparse sets,” joint with Vincent Chan and Malabika Pramanik. The paper is available here, and here is the arXiv link.

Very briefly, the question we consider is the following. Let

for some

In general, the answer is no, even when

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

This paper gives a multidimensional analogue of that result. We define, via conditions on the matrices

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

**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.