New Number Systems Point Geometry Problem Toward a Real Solution

It sounds like a brainteaser. A needle is placed on a table. How much area is needed to turn it so that it points in all directions?

The circle's diameter is the length of the needle. This is completely incorrect. Over the last century, the effort to understand the ways in which it's wrong has revealed a deeply provocative mathematical problem about the nature of the real numbers themselves.

Several proof have recently made some of the most notable progress on the Kakeya conjecture. The results take the original question away from the real numbers and into worlds where lines can be defined by alternative number systems.

The inventiveness has made mathematicians think.

According to Larry Guth, a mathematician at the Massachusetts Institute of Technology who has worked on the problem for more than 15 years, there is a chance that a solution will be found in a few years. I have seen other approaches, but it seems more hopeful.

A Tight Squeeze

The original statement of the problem was made by Sichi Kakeya in 1917. He was curious about the smallest area that was needed to turn a one-dimensional line into a two-dimensional line.

A disk with a diameter equal to the line's length can be used. Smaller shapes can be effective. An equilateral triangle is equal to the length of the line. The desired sweep can be achieved by shifting the line around the triangle, because it is one-dimensional. The Kakeya set is a set of points that allow for thorough pointing.

The smallest area of a Kakeya set was requested by Kakeya. There is no limit to how small it can be. He showed that it is possible to make Kakeya sets that take the equilateral-triangle design to a whole new level. You end up with a lot of spikes in all directions, instead of the three spikes of the triangle.

The author of one of the new proof states that it is a weird looking thing. An area that can be made arbitrary small is what the result is.

The air was taken out of Kakeya's question by Besicovitch's construction. The revised version of the problem would prove much more difficult to solve.

Widespread Emptiness

There are other ways to describe the size of a Kakeya set. In the 1970s there was a question about how efficient the points are in the sets Besicovitch created.

Things can be very close to zero without being zero. That is the crux of the problem.

If you have little squares of fabric, you can position them over a Kakeya set so that the squares cover the set completely.

The extent to which points in a set are arranged in a way that makes them easier or harder to cover is measured by the Hausdorff dimensions. After Besicovitch proved that measuring area alone was not enough to understand the essential properties of the Kakeya sets, mathematicians used these notions of dimensions to explore them. The dimensions of a Kakeya set must be large, according to the Kakeya conjecture. The exact definitions of those two measures of dimensions are technical, but the intuition behind the conjecture is that you need a lot of something.

Imagine you are trying to cram them all into something. Guth wondered how it could be compressed.

Real Problems

In the Kakeya conjecture, points are defined by real numbers, which can have an infinitely long decimal. The Kakeya conjecture is hard to solve because of these real- value coordinates.

It's not clear what the real numbers that create such an obstruction are. The real numbers are continuous, which means that you can't look at them over a long period of time. If you limit yourself to an interval between 1 and 2, the sum of two numbers within that interval will be outside it. The real numbers are uncountably infinite, meaning that no matter how much you zoom in on them, you see the same thing.

Things can be very close to zero without being zero. Joshua Zahl is a professor at the University of British Columbia.

The Kakeya conjecture can be set in smaller number systems due to the difficulty of the real numbers. They might only have the whole number values. The number systems don't look like real numbers, but they have the same basic math properties.

You can ask a slightly modified version of the Kakeya conjecture if you want to know the minimum size of a set of points in one of these number systems. The question was posed by Thomas Wolff in 1996. Since then, mathematicians have looked at it as a way to answer the Kakeya question.

Manik Dhar, an author of two recent papers on the Kakeya case, said that the idea was that it was easier to solve.

Pick a Number

The first thing you need to do is pick a number. Your number system could contain the whole numbers 1 through 9 if you chose 9. Maybe you chose 17 or 25.

It's your choice. The behavior of the number system and the methods used to apply to the Kakeya conjecture can be affected by whether the number is prime or not.

The particular case Wolff had in mind in 1996 was the one Dvir solved in 2008. Number systems called finite fields are used to attack hard problems in mathematics.

Dvir showed that a Kakeya set has the largest possible dimensions in a finite setting. His proof, which was just two pages long, leaned heavily on the fact that when the modulus is prime, any set within the finite number system serves as the solutions to a polynomial equation.

The first major progress on the Kakeya conjecture was represented by Dvir's proof, which made mathematicians hopeful that further advances would be made in the near future.

Nobody showed up. Guth said that people were very excited and that it didn't work.

Dvir returned more than a decade later.

Products of Primes

The Kakeya conjecture for finite number systems in which the modulus is any number that is the product of different primes was solved in November 2020. The number systems had to move beyond the method of the previous one. They turned the problem into a question about matrices.

Points and rows are represented in the matrices. Write a 1 in the corresponding spot if there is a line going in a certain direction. Enter 0 if you don't want to. The matrix is used to show the properties of the lines. The properties of the matrix can now be calculated. The size of the set of lines is related to the matrix'srank.

The rank of these matrices is high, which means the set of lines is large, which means the Kakeya conjecture is true for these number systems.

The result was extended less than a year after it was announced. He showed the Kakeya conjecture for finite number systems in which the modulus is a prime number raised to a power such as 9. The p-adics is an infinite number system that is similar to the real numbers. The mathematicians decided if his methods could be changed to apply to the real numbers.

It became obvious after a few months that they can't be.

There are small differences in how the field of real numbers and p-adic fields behaves.

Two more plot twists have arisen from Arsovski's work. The Kakeya conjecture was proven true by Dhar last October. The local fields of positive characteristic were confirmed by Salvatore in February.

There are many ways to think about this. Maybe the real numbers are next now that mathematicians have proved the validity of the one number system after another. Why haven't mathematicians been able to confirm the Kakeya conjecture for the real numbers given that they've now been able to confirm it in so many other settings?

The explanation might be the most obvious, according to a mathematician.

Guth said that he was no longer confident in the Kakeya theory.

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