Mathematicians Cross the Line to Get to the Point

Imagine an endless piece of blank paper covered with a smattering of lines pointing every which way. A gust of wind comes and sprinkles dust on top of the paper — in effect covering the lines with points. Say a helpful mathematician tells you how much dust covers any one line. Based on that one piece of information, can you figure out how much dust is there in total?

While there isn’t enough information to give an exact answer, in 1999 Thomas Wolff, a mathematician at the California Institute of Technology, published a guess about the minimum amount of dust that must be covering the paper. This came to be called the Furstenberg set conjecture, after Hillel Furstenberg, whom Wolff credited with coming up with the problem.

For two decades, the conjecture stumped anyone who tried to prove it. The best progress came in 2003, when Jean Bourgain, a legendarily astute mathematician, inched a tiny bit closer in one special case. But this August, Kevin Ren, a second-year graduate student at Princeton University, and Hong Wang, a professor at New York University, posted a proof of the full 1999 conjecture on the preprint site arxiv.org. Even mathematicians directly involved in the line of research that had led to Ren and Wang’s solution were taken aback by their announcement. “It was a huge surprise,” said Pablo Shmerkin, a mathematician at the University of British Columbia and a collaborator of Wang’s.

To understand the Furstenberg set problem, it first helps to understand a number called the Hausdorff dimension. For everyday objects, the Hausdorff dimension agrees with ordinary intuition about dimensions: An individual point is zero-dimensional; a line is one-dimensional, the endless blank paper is two-dimensional. But the Hausdorff dimension is a little more nuanced. It describes how concentrated sets of points are, even when they aren’t easily described as shapes.

For example, think of a line segment that goes from zero to 1. Take away the middle third — everything between 1/3 and 2/3. Then take away the middle third of each remaining third — everything between 1/9 and 2/9 and between 7/9 and 8/9. Keep repeating this process, and from a certain perspective, you’ll have nothing left: The resulting set will cover so little of the original line segment that its length will be zero. But it is, in both an intuitive and a mathematical sense, “bigger” than just a single point. Its Hausdorff dimension is about 0.6.

Or consider a curve like the one below:

It is in a sense “bigger” than a straight line, but smaller than a two-dimensional object. Its Hausdorff dimension is about 1.26.

Return to Wolff’s endless piece of paper. The number the helpful mathematician gives you is the minimum Hausdorff dimension of all the dust you’ll see if you examine any one particular line. Call this number s. Wolff showed that the Hausdorff dimension of all the dust had to be at least s + ½ or 2s, whichever is bigger. (If s is less than ½, then s + ½ is bigger; if it’s more, then 2s is.) Though he included a proof, he didn’t claim priority, writing that “it is unlikely that the author was the first to observe these bounds.” He wasn’t sure who had first proved them — possibly it was Furstenberg.

Wolff suspected the limit was higher than he could prove. He thought the dimension must be at least $latex \frac{3s+1}{2}$, which is always bigger than either of the other options. That suspicion became known as the Furstenberg set conjecture.

Ren first encountered the conjecture in 2020 as an undergraduate at the Massachusetts Institute of Technology. He was ambitious and thought he would try and prove the problem. But after reading Bourgain’s paper from nearly two decades before, he got discouraged. He couldn’t understand the paper. “It must have been a very beautiful proof, but at that time I was like, ‘Oh, no, I cannot read this,’” Ren said.

But he never completely let go of the question. This past June, he found a new paper by Shmerkin and Tuomas Orponen of the University of Jyväskylä in Finland that proved a special case of the conjecture. And he knew of a 2019 paper co-authored by Larry Guth, a mathematician at MIT who had advised him as an undergraduate, which had proved a different special case. Ren thought he could combine the two special cases in a way that would make a general proof.

The 2019 paper had proved the conjecture for lines that are arranged in clumps — that is, lines that look spaced out from far away but turn out to be a dense bundle if you zoom in. The June paper covered the opposite case — regular patterns of lines whose dimension looks the same no matter how much you zoom in or out. “I started thinking about it for the next three weeks,” he said. As he did chores, he said, he kept imagining sets of lines passing through points.

Soon, his efforts paid off. He realized that if you zoomed in and out, you could show that any set of lines was either clumpy or regular. He was able to cobble together a proof that worked no matter what kind of set you had.

Ren contacted Guth, who responded with surprising news: Wang, one of his co-authors on the 2019 paper, also had a proof. Wang and Ren got in touch and discovered that their strategies were eerily similar. The two merged their arguments and posted their proof online on August 17.

Nets Katz, a professor at Rice University who has also worked on the conjecture, cautions that Ren and Wang’s paper is still a preprint and hasn’t been thoroughly reviewed by him or other experts. “We’re not an old boys’ network. If we were, I would just tell you Hong is brilliant, because she is,” he said. Shmerkin puts the probability of correctness at 95%. But Ren feels comforted by the fact that he and Wang separately thought of the same strategy. “This draft is already less technical than our very initial draft,” he said. “I hope when we publish it, the final journal version will be as elegant as a work of art.”