![]() ![]() Developed by the Hungarian number theorist Imre Ruzsa and Cilleruelo, this approach yields bigger, denser Sidon sets than the probabilistic method, which Pilatte needed to create a basis of low order that also obeyed the Sidon property. So Pilatte took a different tack, turning instead to a procedure that uses the logarithms of prime numbers as the building blocks of Sidon sets. “You can get a basis of order 4 by using probabilistic methods, but you can’t get a basis of order 3,” he said. Pilatte realized that the probabilistic method had been pushed as far as it could go. These findings were obtained using variations of a probabilistic method pioneered by Erdős that involves generating a random set of integers and tweaking it slightly to create a set that satisfies both properties. Two years later, the Spanish mathematician Javier Cilleruelo took these results a step further by proving that it is possible to construct a Sidon set that is an asymptotic basis of order 3 + ε , meaning that any sufficiently large integer N can be written as the sum of four members of the Sidon set, with one of them smaller than N ε for arbitrarily small positive ε. In 2010, the Hungarian mathematician Sándor Kiss showed that a Sidon set can be an asymptotic basis of order 5 -meaning that any sufficiently large integer can be written as the sum of at most five elements of the set - and in 2013 Kiss and two of his colleagues proved the conjecture for an asymptotic basis of order 4. Some progress had been made before Pilatte took up the challenge. “But there was a difficulty with the techniques we were using.” “People believed this should be true,” said Pilatte’s adviser James Maynard. It was generally believed that an asymptotic basis of order 3 could be constructed from a Sidon set, but proving this was another matter. (The odd numbers, for example, form a basis of order 2.) As Pilatte explained, this is so simple to show that mathematicians didn’t bother to write it down: “That order 2 is impossible was probably known much earlier than it was explicitly written in the literature.” He explained that this is because “Sidon sequences cannot exceed a certain density, while asymptotic bases of order 2 are always denser than that threshold, so the two properties cannot hold at once.” It has long been known that a Sidon set cannot be an asymptotic basis of order 2, where any integer can be expressed as the sum of at most two numbers. ![]() For example, add together any two numbers in $.) Sidon sets, in other words, have to be sparse. The first is that no two pairs of numbers in the set add up to the same total. Pilatte proved that it is possible to create a set - a collection of numbers - that satisfies two apparently incompatible properties. Such is the case with a new proof posted online in March by Cédric Pilatte, a first-year graduate student at the University of Oxford. Mathematicians rejoice when they prove that seemingly impossible things exist. ![]()
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