# Finite configurations in sparse sets

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 $E \subseteq \mathbb{R}^n$ be a closed set of Hausdorff dimension $\alpha$. Given a system of $n \times (m-n)$ matrices $B_1, ... ,B_k$ for some $m \geq n$, must $E$ contain a “non-trivial” k-point configuration

$(1)\ \ \ x + B_1 y,\ ...,\ x + B_k y$

for some $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^{m-n}$?

In general, the answer is no, even when $\alpha=n$. For instance, Keleti has constructed 1-dimensional subsets of $\mathbb{R}$ that do not contain a similar copy of any given triple of points $(x,y,z)$ (in fact, his construction can avoid all similar copies of all such triples from a given sequence), as well as 1-dimensional subsets of $\mathbb{R}$ that do not contain any non-trivial “parallelograms” $\{x, x+y, x+x, x+y+z\}$. In $\mathbb{R}^2$, given any three distinct points $a,b,c$, Maga has constructed examples of sets of dimension 2 that do not contain any similar copy of the triangle $abc$; he also constructed sets of full dimension in $\mathbb{R}^n$, for any $n\geq 2$, 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 $E\subset \mathbb{R}$ has dimension close enough to 1, and if it also supports a measure obeying appropriate dimensionality and Fourier decay estimates, then $E$ must contain a non-trivial 3-term arithmetic progression. The same proof applies to any other configuration $x,y,z$, 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 $B_j$, 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 $B_j$, if $E\subset \mathbb{R}^n$ has dimension close enough to $n$, and if it supports a measure with dimensionality and Fourier decay conditions similar to those in my paper with Pramanik, then $E$ 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 $B_j$ 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 $\mathbb{R}^2$. Let $a,b,c$ be three distinct points in the plane. Suppose that $E\subset\mathbb{R}^2$ satisfies the assumptions of our main theorem. Then $E$ must contain three distinct points $x,y,z$ such that the triangle $\triangle xyz$ is a similar (possibly rotated) copy of the triangle $\triangle abc$.
• Colinear triples in $\mathbb{R}^n$. Let $a,b,c$ be three distinct colinear points in $\mathbb{R}^n$. Assume that $E\subset\mathbb{R}^n$ satisfies the assumptions of our main theorem. Then $E$ must contain three distinct points $x,y,z$ that form a similar image of the triple $a,b,c$.
• Parallelograms in $\mathbb{R}^n$. Assume that $E\subset\mathbb{R}^n$ satisfies the assumptions of our main theorem. Then $E$ contains a parallelogram $\{x,x+y,x+z,x+y+z\}$, where the four points are all distinct.