AlphaEvolve rediscovered best-known solutions for most of 67 tested math problems and found improved solutions in several cases using LLM-guided evolutionary search.
Control and its applications in additive combinatorics.arXiv:2501.09470, 2025
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For any A subset of reals with Hausdorff dimension s in (0,1/2], either upper box dim(AA) or lower box dim(A+A) is at least 29s/23 (or 33s/26 for differences).
Adapts known construction to prove existence of c>0 and large finite A subset R with |AA+A+A| << |A|^{2-c}, plus corollaries for other sum-product expressions.
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Mathematical exploration and discovery at scale
AlphaEvolve rediscovered best-known solutions for most of 67 tested math problems and found improved solutions in several cases using LLM-guided evolutionary search.
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A note on the sum-product problem for fractal sets
For any A subset of reals with Hausdorff dimension s in (0,1/2], either upper box dim(AA) or lower box dim(A+A) is at least 29s/23 (or 33s/26 for differences).
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More sum-product type counterexamples: products with shifts and $AA+A$
Adapts known construction to prove existence of c>0 and large finite A subset R with |AA+A+A| << |A|^{2-c}, plus corollaries for other sum-product expressions.