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acid andy [soylentnews.org] writes:

For 30 years, mathematicians wondered if you could have an infinite set of numbers where each pair of numbers adds up to a unique value, and have those values each be fairly large.

In March, a graduate student [Cédric Pilatte [arxiv.org]] from Oxford University finally solved the problem by turning to an unlikely solution: geometry.

In 1993, Hungarian mathematician [real.mtak.hu] Paul Erdős—one of the most prolific mathematicians of the 20th century—posed a question with two components seemingly at odds with one another: Could a Sidon set be an “asymptotic basis of order three?”

[...] Named after another Hungarian mathematician, Simon Sidon, these sets are basically a collection of numbers where no two numbers in the set add up to the same integer. For example, in the simple Sidon set (1, 3, 5, 11), when any of the two numbers in the set are added together, they equal a unique number. Constructing a Sidon set with only four numbers is extremely easy, but as the set increases in size, it just gets harder and harder. As soon as two sums are the same, the collection of numbers is no longer considered a Sidon set.

The second element of Erdős’ problem—that scary-sounding “asymptotic basis of order three” part—means that:

1. a set must be infinitely large
2. any large enough integer can be written as the result of adding together at most 3 numbers in the set.

So, this 30-year-old conundrum centered on whether or not these two elements could exist in the same set of numbers. For decades, the answer seemed to be no.

[...]So how did Pilatte get a mathematically square peg to fit a seemingly round hole? He took an unconventional approach and turned to geometry [popularmechanics.com] rather than the probabilistic method championed by Erdős and what’s called additive number theory. Pilatte replaced numbers with polynomials and made use of the recent work of Columbia University [arxiv.org] mathematicians. Combining these ideas, Pilatte successfully created a Sidon set dense enough and random enough to finally solve Erdős’s original problem.

https://arxiv.org/abs/2303.09659v1

Original Submission