To explain quickly what this is. If you take a set of numbers and add all of them pairwise you get a new set. Similarly if you multiply them pairwise you get a new set. Then we can make certain statements about the size of the original set vs the new one.
What the Sum-Product conjecture says is that you can intentionally built a set to create more intersections in the sum set, or you can do that with the product set, but not both. If it weren't for intersections any pairwise operation would result in an ~N^2 size in the resulting set. What the Sum-Product conjecture says specifically is that max(|sumset(A)|,|prodset(A)|) approaches |A|^2 as A gets very large.
So the disproof is that mathemiticians found a way to create a set that encourages intersections in both the sum set and product set at the same time.
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To explain quickly what this is. If you take a set of numbers and add all of them pairwise you get a new set. Similarly if you multiply them pairwise you get a new set. Then we can make certain statements about the size of the original set vs the new one.
What the Sum-Product conjecture says is that you can intentionally built a set to create more intersections in the sum set, or you can do that with the product set, but not both. If it weren't for intersections any pairwise operation would result in an ~N^2 size in the resulting set. What the Sum-Product conjecture says specifically is that max(|sumset(A)|,|prodset(A)|) approaches |A|^2 as A gets very large.
So the disproof is that mathemiticians found a way to create a set that encourages intersections in both the sum set and product set at the same time.