On sets containing a unit distance in every direction

On sets containing a unit distance in every direction, Discrete Analysis 2021:5, 13 pp.

A Kakeya set in $\mathbb R^d$ is a subset $A\subset\mathbb R^d$ that contains a line in every direction. Besicovitch famously proved that a Kakeya set in $\mathbb R^2$ can have measure zero, also solving the closely related Kakeya needle problem by showing that there are sets of arbitrarily small area inside which a needle can be turned round so that it is facing in the opposite direction.

One can also consider more refined measures of smallness than Lebesgue measure. For example, it can be shown that the Hausdorff dimension of a Kakeya set in $\mathbb R^2$ must be 2, so in that sense planar Kakeya sets must be large. It is conjectured that Kakeya sets in $\mathbb R^d$ must have Hausdorff dimension $d$, but for $d\geq 3$ this is a major open problem.

Another major open problem is Erdős’s unit distance problem, which is the following question. Let $\{x_1,\dots,x_n\}$ be a set of $n$ points in the plane. How many pairs $\{i,j\}$ can there be such that the distance between $x_i$ and $x_j$ is 1? It is conjectured that the maximum is $n^{1+o(1)}$, but the best known upper bound is $n^{4/3+o(1)}$. Note that a proof of this conjecture would imply that the number of distinct distances $d(x_i,x_j)$ is always $n^{1-o(1)}$, which itself was a famous open problem until it was solved by Larry Guth and Nets Katz in 2010.

This paper concerns a kind of cross between the Kakeya problem and the Erdős unit distance problem. The basic question it treats is to determine how small a subset $A\subset\mathbb R^d$ can be if it contains a unit distance in every direction. That is, instead of asking for a line in every direction, it asks for a pair of point along that line at distance 1. Note that the condition on $A$ is equivalent to the condition that the difference set $A-A$ contains the unit sphere of $\mathbb R^d$. The authors call such sets Kakeya dipole sets.

Since this is a weaker condition than the condition on Kakeya sets, we know immediately that $A$ can have measure zero, which is obvious anyway since we could take $A$ to be a sphere of radius 1/2, or a sphere of radius 1 together with its centre. More interestingly, it can be shown by a Baire category argument that there are Kakeya dipole sets of Hausdorff dimension 0.

This might seem to be the end of the matter, but it isn’t, because Hausdorff dimension is not the only natural definition of dimension. One of the notions considered in this paper is that of box dimension. Given a set $A\subset\mathbb R^d$, let $N_\epsilon(A)$ be the number of cubes of sidelength $\epsilon$ needed to cover $A$. If $A$ is a “nice” set of dimension $k$ (e.g. a smooth $k$-dimensional submanifold of $\mathbb R^d$) and is bounded, then the growth rate of $N_\epsilon(A)$ will be of order $\epsilon^{-k}$. One can then turn this observation into a definition of dimension, by saying that if $N_\epsilon(A)$ grows at a rate of order $\epsilon^{-k}$, then $A$ has dimension $k$. Note that $k$ does not have to be an integer: for example, the box dimension of the Koch snowflake is $\log 4/\log 3$.

More precisely, the upper box dimension of $A$ is $\lim\sup\log N_\epsilon(A)/\log(1/\epsilon)$ and the lower box dimension of $A$ is $\lim\inf\log N_\epsilon(A)/\log(1/\epsilon)$, where the limits are taken as $\epsilon\to 0$. If these two coincide, then their common value is the box dimension of $A$.

Hausdorff dimension also involves covering a set with small cubes. Roughly speaking, the difference between the box dimension and the Hausdorff dimension is that with the Hausdorff dimension the cubes do not have to be the same size – they just have to have sidelength at most $\epsilon$, with an appropriately smaller “cost” if smaller boxes are used. This additional flexibility implies that Hausdorff dimension is smaller than or equal to the box dimension, which raises the question of what one can say about the (lower or upper) box dimension of a Kakeya dipole set.

A trivial argument shows that the lower box dimension must be at least $(d-1)/2$ – it can be checked easily that the dimension of $A-A$ is at most the dimension of $A\times A$, which is equal to twice the dimension of $A$. Therefore, if $A-A$ contains the unit sphere, $A$ must have at least half the dimension of the unit sphere. One of the main results of this paper is to improve this trivial bound when $d=2$ from $1/2$ to $4/7$.

The authors also prove similar results for the Assouad dimension of Kakeya dipole sets. This is a more local definition: it is the infimum of all $s$ such that the intersection of $A$ with a ball of radius $R$ can be covered with order $(R/\epsilon)^s$ balls of radius $\epsilon$. It is at least as large as the upper box dimension, and the bounds obtained in the paper are larger.

In addition to these lower bounds, the authors provide constructions that give upper bounds in all dimensions. In particular, they construct a planar dipole Kakeya set with lower box dimension at most 2/3. Closing the gap between this and the lower bound above is an appealing open problem.