Computer scientists made progress on a decades-old puzzle in a subfield of mathematics known as combinatorics.
Consider this sequence of numbers: 5, 7, 9. Can you spot the pattern? Here’s another with the same pattern: 15, 19, 23. One more: 232, 235, 238.
Kelley and Meka were instead investigating abstract games in computer science. The pair sought a mathematical tool that might help them understand the best way to win a particular type of game over and over again. “I’m super-interested in a collection of techniques that fall under this umbrella called structure versus randomness,” Kelley says. Some of the earliest progress on arithmetic progressions relied on such techniques, which is what led Kelley and Meka to dive into the topic.
Potential applications aside, researchers who study arithmetic progressions — or other facets of purely theoretical mathematics — are often motivated more by sheer curiosity than any practical payoff. The fact that questions about such simple patterns and when they appear remain largely unanswered is, for many, reason enough to pursue them. Let’s take a moment to get our hands on some sets of numbers and the arithmetic progressions those sets contain, starting with the.
A sparse enough set of numbers can have gaps arranged in ways that manage to avoid arithmetic progressions. Too dense, though, and the set can’t avoid having gaps that match up. In the 20th century, mathematicians settled on a way to measure that density. They are now looking for the density above which arithmetic progressions must appear.
Let’s return to the powers of 10, whose density decreases across the number line. As you go out farther and farther, the proportion of whole numbers that are powers of 10 approaches zero — so the set has an asymptotic density of zero. Other sets have a positive asymptotic density, and some never settle down into an asymptotic density at all.
It’s not quite a hop, skip and a jump from parallel repetition of games to arithmetic progressions, but Kelley and Meka got there fairly quickly. “Maybe in a month we were at the 3-AP problem,” Meka says. Previous research on parallel repetition of games had used structure versus randomness arguments. Because Roth’s work on arithmetic progressions was the first to use such a technique, Kelley and Meka were interested in that work in its original habitat.
Roth’s initial proof found an upper limit to where the limbo bar must be. He showed that any set whose density approaches zero at a rate similar to or slower than the expression 1/log) must contain at least one arithmetic progression. Log means to take the logarithm, and remember that N is the number chosen as the arbitrary cutoff in an infinite set. We’re considering what happens as N increases.
Kelley and Meka pushed the upper limit down dramatically. Their result was a rate that goes to zero at approximately the same rate as 1/e. That formula looks eerily similar to Behrend’s lower limit. For the first time ever, the upper and lower limits are within shooting distance of each other. Closing that gap would reveal the specific location of the limbo bar and thus give a clear answer to which sets must contain at least one 3-AP.
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