On distance sets, box-counting and Ahlfors-regular sets

On distance sets, box-counting and Ahlfors-regular sets, Discrete Analysis 2017:9, 22 pp.

A well-known problem of Falconer, a sort of continuous analogue of the Erdős distinct-distance problem, asks how large the Hausdorff dimension of a Borel subset of $\mathbb R^d$ needs to be before the set of distances between points of the subset becomes large, where in some versions of the problem one asks for it to have positive measure and in others for it to have Hausdorff dimension 1. Falconer conjectured that if the dimension of the set is at least $d/2$, then the dimension of the set of distances is 1, and proved that this would be sharp. The example that shows this is based on the fact that lattices do not give rise to too many distinct distances. Define the set $S_{a,\delta}$ to be the set of all points $(x,y)$ such that both $x$ and $y$ are within $\delta$ of a multiple of $a$. If $s < d/2$, one takes a rapidly decreasing sequence of $a_i$, and chooses suitable $\delta_i$ to make the set have dimension $s$. The set of distances can then be shown quite easily to have dimension less than 1.

In the plane, this conjecture would tell us that the dimension needed to guarantee a distance set of dimension 1 is itself 1. To date, this problem is open, with the best result being that the conclusion holds for any dimension greater than 4/3.

A considerably stronger conjecture, which still appears to be open, is that if $A$ is a planar set of dimension at least 1, then there exists $x\in A$ such that the set $\{d(x,y):y\in A\}$ has dimension 1. This set is called a pinned distance set. This is known if $A$ has dimension greater than 3/2.

This paper concerns a variant of the problem where one makes quite a strong assumption about the set $A$ and aims to prove a stronger conclusion. To understand the assumption, consider the following property of the middle-thirds Cantor set in $[0,1]$. There is a natural probability measure on this set: for each ternary interval of length $3^{-k}$ with an interior that intersects the set, the measure of the intersection is $2^{-k}$. It is not hard to check that if $x$ is a point of the Cantor set and $0 < r \leq 1$, then the measure of the interval $[x-r,x+r]$ will be between $C^{-1}r^{2/3}$ and $Cr^{2/3}$ for some absolute constant $C>1$. A subset $A$ of $\mathbb R^d$ is said to be $s$-Ahlfors regular with constant $C$ if it has a property of this kind, namely that there is a measure $\mu$ supported on $A$ such that for every $x\in A$ and every $r$ the ball of radius $r$ about $x$ has $\mu$-measure between $C^{-1}r^s$ and $Cr^s$. It can be shown that an $s$-Ahlfors regular set has Hausdorff dimension $s$, but as the name suggests, this is true in a much more “regular” way than has to be the case for an arbitrary set of Hausdorff dimension $s$.

In a recent breakthrough, Tuomas Orponen showed that an $s$-Ahlfors regular set $A$ in the plane with $s>1$ has a distance set of packing dimension 1. The packing dimension is greater than or equal to the Hausdorff dimension, so this does not prove Falconer’s conjecture for Ahlfors regular sets, but it is a significant step in the right direction, and a very interesting new approach to the problem.

This paper proves a result that strengthens Orponen’s in several ways: assuming a slightly weaker property than Ahlfors regularity, it proves that the modified lower box-counting dimension has to be 1 (which is a stronger statement, since the modified lower box-counting dimension is less than or equal to the packing dimension – for the definitions, see the paper), it proves this not just for the full distance set but for many pinned distance sets, and it obtains results for the set of distances $\{d(x,y):x\in A, y\in B\}$ for sets $A$ and $B$ that do not have to be equal. Furthermore, discretized versions of the box-counting estimates are proved that imply non-trivial statements about finite sets of points that satisfy a discrete version of Ahlfors regularity. Of additional interest is the method of proof, which, while using several of Orponen’s ideas, also introduces new ones in order to obtain the stronger results. In particular, the proof makes use of the notion of a CP-process from ergodic theory. In the words of the author: “Very roughly speaking, a CP-process is a measure-valued dynamical system which consists in zooming in dyadically towards a typical point of the measure.”

In the video below, the author provides some context for this result: the result itself is discussed briefly at the end (starting at about 52:53).