In a paper (Orponen and Shmerkin 2023), recently accepted for publication in the Journal of the American Mathematical Society, Orponen and Shmerkin gave a positive solution to the ABC sum-product problem.
Sum-product theorems over the integers, discussed in this previous blog post, assert that for a finite set , the sumset and the product set cannot both be small. In that setting, the relevant sets are finite, and their size is measured by cardinality. In 2003, Bourgain (Bourgain 2003) established an analogue for infinite subsets of the real line, where “size” is quantified using -covering numbers. Let denote the ball in centred at with radius . For and any set , write for the minimum number of balls of radius needed to cover . Bourgain proved that for any , there exists such that, for all sufficiently small , the following holds: if satisfies and then
This theorem has several striking consequences. For instance, it implies that if are Borel sets with and , then there exists such that In 2022, Orponen (Orponen 2022) conjectured that should remain valid in the more general regime where and . He also showed that, if true, this statement would be sharp. This became known as the ABC sum-product problem. In 2023, Orponen and Shmerkin posted a preprint (Orponen and Shmerkin 2023) confirming the conjecture.
In other words, the theorem shows that the threshold predicted by Orponen is indeed the correct one: as soon as has dimension exceeding by a positive amount, one can find a parameter for which the set has strictly larger dimension than . This gives a definitive resolution of the ABC sum-product problem and marks a major advance in the theory of sum-product phenomena over real numbers.