It’s one thing to turn a cartwheel in an open field. It’s another to manage it in a tight space like a bathtub. And that, in a way, is the spirit of one of the most important results in number theory over the past two decades.
The result has to do with the “sum-product problem,” which I wrote about last week. It asks you to take any set of numbers and arrange them in a square grid, then fill in the grid with either the sums or the products of the crosswise pairs.
The sum-product problem states that the number of distinct sums or products will always be close to N2 (where N stands for the number of numbers used to make your grid).
The sum-product problem that I wrote about uses any set of real numbers to generate the grid. You can also restrict the problem to use unique number systems that are smaller and more constrained than the reals. These self-contained number systems are called “finite fields.”
In mathematics, a “field” is any number system in which you can carry out the four basic operations of arithmetic: addition, subtraction, multiplication and division. The real numbers form a field. You can perform those operations on any two real numbers and the outcome is a third real number. Or, put another way, the arithmetic of the real numbers never yields a number outside the real numbers.
The integers — all the positive and negative counting numbers — don’t form a field. Yes, you can add, subtract and multiply any two integers to produce a third integer. But divide 3 by 2 and you’ll get 1½, which isn’t an integer.
A “finite” field is a number system in which the number of numbers is finite. There are different kinds of finite fields, but the simplest one has to do with what’s called “modular” or “clock” arithmetic. In modular arithmetic, when you get to the end of your finite list of numbers, you just loop back to the beginning, as if you were counting numbers around the face of a clock. For example, if you go to a party at 7 p.m. and get back home six hours later, you’ll return at 1 a.m. More formally, 7 plus 6 in the base-12 modular number system equals 1.
(The 12 numbers on the clock don’t actually form a field, and the reason why has to do with one of the most critical features of number theory: Modular number systems form fields only when they’re comprised of a prime number of elements. In modular number systems with a nonprime number of elements, like the 12-digit clock, you end up in weird situations where the product of two nonzero numbers is zero. For example, 6 × 4 = 24, which equals 0 in the base-12 number system. This leads to other consequences that cause division to break down. But in a modular number system with a prime number of elements, two nonzero numbers never multiply to zero.)
Finite fields have been the setting for many celebrated results in mathematics. As self-contained arithmetic worlds, they contain a rich structure that mathematicians can exploit to solve problems related to everything from prime numbers to patterns in the solutions to polynomial equations.
In 2003, the mathematicians Jean Bourgain, Nets Katz and Terry Tao became the first mathematicians to make any progress on the sum-product problem over finite fields. They proved that either the number of distinct sums or the number of distinct products must be at least ever-so-slightly larger than the size of the set used to generate the sum and product grids. The statement was modest in size but momentous in significance.