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

Abstract

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 $s 1$. 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](https://wiki.helsinki.fi/display/mathstatHenkilokunta/Orponen%2C+Tuomas) 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).