The Falconer distance conjecture

The explanation

Draw a square and cut it into a 4 by 4 grid of 16 small squares. Keep 5 of them and throw the rest away. Now repeat the same step inside each square you kept, forever. After 5 steps you hold 3125 tiny squares, each 1/1024 of the side. Together they cover about 0.3 percent of the original area. In the limit the area is zero. What remains is a fractal dust of points.

Now measure every distance between two points of that dust. The distances all lie between 0 and about 1.414, the diagonal. The question is whether they have positive total length, even if they leave gaps. A finite set of points gives only finitely many distances, so the length is zero. A full square gives every distance from 0 to 1.414. The dust sits in between.

The right yardstick is Hausdorff dimension. It is a way to give a set a size between whole numbers. A line has dimension 1 and a filled square has dimension 2. Our dust multiplies its pieces by 5 while shrinking them by 4 at each step. That gives it dimension log 5 / log 4, about 1.16. The plane has dimension 2, so half of that is 1. The dust is above that halfway mark.

In 1985, K. J. Falconer asked what dimension forces the distances to have positive length. Positive length here means positive Lebesgue measure, the usual notion of total length on a line. He showed that dimension above (d+1)/2 is enough, where d is the dimension of the space. In the plane that is 1.5. He also gave lattice-type examples at exactly d/2. These explain why the conjecture asks for dimension strictly greater than d/2.

Later work pushed the planar threshold down. Thomas Wolff reached 4/3 in 1999. In 2020, Guth, Iosevich, Ou and Wang reached 5/4 with a stronger, pinned conclusion. From some single point of the set, the distances to the others already have positive length. Our dust, at about 1.16, sits below both planar thresholds. In three dimensions, Du, Guth, Ou, Wang, Wilson and Zhang reached 9/5 for distances between all pairs. A 2024 preprint by Du, Ou, Ren and Zhang gave the pinned conclusion above about 1.714. The target there is 1.5. Results near the halfway mark needed extra conditions. Liu's planar result needs the set's Hausdorff dimension to equal its packing dimension, a second way to measure size. Shmerkin and Wang worked at exactly d/2, for sets with both dimensions equal to d/2. They showed the distances have dimension 1, which is weaker than positive length.

This paper proves the full conjecture in every dimension d of 2 or more. Take any compact set, meaning closed and bounded, in d-dimensional space. If its Hausdorff dimension is above d/2, its distances have positive total lengthSource 1. In the plane, any such set above dimension 1 qualifies, including the dust. The theorem asks for no regularity at allSource 2. A set may be as patchy and uneven as it likes.

The main obstacle shows up in higher dimensions. Stand at one point of the set and look at the other points. Note the direction to each one. Those directions form a set on a sphere. For a set of dimension just above d/2, the directions can be too thin to have a densitySource 3. A density here means a smooth spread over the sphere. So the proof cannot assume one. It instead builds a weaker guarantee: no small patch of directions holds too much of the setSource 4. It then shows that this weaker guarantee is what the distance estimate actually needs.

The result has clear edges. It says nothing at dimension exactly d/2Source 5. It is about all distances between pairs of points. It does not claim that the distances measured from one single point have positive lengthSource 6. It also gives positive length, which is stronger than saying the distances have dimension 1Source 7.

How sure is this? A computer proof checker called Lean was used. Its scope note says the formal proof covers the main theorem in every dimension d of 2 or more. The note links two formal statements: this all-dimension result and the earlier planar case. Specialists have not published a review yet.

What it doesn't show. This does not cover sets of dimension exactly d/2, or distances measured from one single point.