Shulou(Shulou.com)11/24 Report--

Imagine a huge blank piece of paper in front of you, with many lines drawn on it, each pointing in a different direction.

A sudden gust of wind blew and some dust fell on the paper.

Then a helpful mathematician appears and tells you how much dust is on a certain line.

Can you calculate the total amount of dust on all the lines based on this information?

The above mathematical problem comes from the Furstenberg set conjecture (the Furstenberg set conjecture).

It was born in 1999, and it has been 24 years since--

Although it is still young in the history of mathematics, it does not look simple.

But the good news:

Princeton sophomore Kevin Ren and Professor Wang Hong of New York University have fully proved it.

And interestingly, the two had never met before, and it was in their respective studies that they "happened to coincide" with the same method before they collaborated to publish the results.

Chinese teachers and students solve the mathematical conjecture born in 1999 to solve this conjecture, we must first master the concept of Hausdorff dimension.

In popular terms, the mathematical model closest to this idea is the topological dimension.

For everyday objects, such as straight lines and rectangles, their topological dimensions (and Hausdorff dimensions) must be integers (1 and 2, respectively).

However, this concept has encountered difficulties in describing some irregular sets such as fractals, and Hausdorff dimension is an appropriate tool to describe such sets.

In these cases, it may be a non-integral rational or irrational number.

For example, the Koch curve (its four iterations below), each consisting of four lines of the same shape as its own scale at 1:3, has a Hausdorf dimension of about 1.26.

In a sense, this number means that it is larger than a straight line, but smaller than a two-dimensional object.

Let's go back to the beginning of the question.

It was actually Thomas Thomas Wolff, a mathematician at the California Institute of Technology, who first raised the question.

He also gave a guess of the minimum amount of dust.

According to the number given to you by the mathematician in the title, we can get the minimum Hausdorff dimension of all the dust seen on a particular line.

We call it s.

Wolf proved that the Hausdorff dimension of all dust must be at least s + half or 2s, whichever is the greater.

But he said he only provided proof and was not sure who came to the conclusion first.

And he himself suspects that the ultimate limit may be higher than that:

At least (3s+1) / 2.

This suspicion is also named "Furstenberg conjecture" in the field of mathematics.

In 2020, Kevin Ren, an undergraduate at MIT, came across this conjecture for the first time.

After reading the paper of mathematician Jean Bourgain, he was still confused.

The mathematician made some progress on a special case in 2003. )

But Kevin Ren has never given up on this issue.

In June this year, he found that a new paper published by Finland at the University of Wesley proved another special case of the conjecture.

Coupled with the special case proved in a paper co-authored by MIT mathematician Larry Guth in 2019, it made him feel:

If you combine these two special cases in some way, can you give a general proof?

Specifically, the 2019 study confirmed the conjecture that when we pull in lines that are far apart from a distance, they actually take the form of a "dense cluster" (a dense bundle).

The June paper gives the opposite story:

The dimensions of regular lines look the same no matter how much they are zoomed in or out.

Kevin Ren spent the next three weeks thinking about this, and his brain kept imagining "thread sets through points" while doing housework.

Soon, his inspiration came.

He realized that if we zoom in or out of a set of lines, the whole can only look like a clumpy or a regular regular.

Based on this, he was able to piece together a proof that is valid for any kind of set.

Excited Kevin Ren quickly contacted Larry Gus (he coached Kevin Ren at MIT), but Gus told him:

Professor Wang Hong of New York University, one of the co-authors of his 2019 paper, also proved it.

But the amazing thing is, when they got in touch with each other, they found out:

Their ideas happen to coincide, and the strategies used are strikingly similar.

Such being the case, they chose to merge their arguments and publish a paper together.

Finally, Professor Nets Katz from Rice University (who was also involved in the study of the conjecture) commented:

At present, the papers of Kevin Ren and Wang Hong are still preprinted and have not been fully peer-reviewed.

But I estimate that the correct rate is 95%.

The author introduces Kevin Ren, a sophomore at Princeton.

The research direction is Fourier analysis and its application in geometric measure theory and metric geometry problems.

He is an undergraduate (2018-2022) from MIT with degrees in mathematics and physics.

Wang Hong graduated from Dr. MIT in 2019.

He is currently an associate professor of mathematics at New York University at Kurant and previously served as an assistant professor at UCLA for two years.

Her research direction is also Fourier analysis and related problems.

Paper address:

Https://arxiv.org/abs/2308.08819

Reference link:

Https://www.quantamagazine.org/mathematicians-cross-the-line-to-get-to-the-point-20230925/

This article is from the official account of Wechat: quantum bit (ID:QbitAI), author: Fengcai

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