'Once in a Century' Proof Settles Math's Kakeya Conjecture

Even the Kakeya set that overlaps the most has to take up some space, Fefferman found. That minimum volume depends on how thick the tubes are. Mathematicians quantify the relationship between the tubes’ thickness and the volume of the set using a number called the Minkowski dimension. The smaller the Minkowski dimension, the more you can reduce the set’s volume by thinning the tubes slightly. The three-dimensional Kakeya conjecture says that a set’s Minkowski dimension must be three. This constitutes a very weak relationship — if you halve the tubes’ thickness, for instance, you will only remove a sliver of the volume at most. Yet even that mild constraint turned out to be nearly impossible to prove. Baby Steps In 2022, five decades after the modern Kakeya conjecture was formulated, Wang and Zahl took a significant step forward. Following a program that Katz and Terence Tao had laid out back in 2014, they examined a pesky class of Kakeya sets. Their proof showed that every set in t ... Read full article.