Sum-difference problem III

AlphaEvolve improved the best known bound; the record has since been surpassed. The problem remains open.

Problem
Problem 44 in the AlphaEvolve repository: Sum-difference problem III
Credit
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
Checked
Construction can be checked directlyPreprint (arXiv, 2025)
Timeline
Paper submitted 2025-11-03
Sources
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
PARTIALImproved bound · new construction

The problem

Let CC be the supremum of all constants such that there exist arbitrarily large finite sets of integers A,BA,B with ∣A+B∣≲∣A∣|A+B| \lesssim |A| and ∣A−B∣≳∣A∣C|A-B| \gtrsim |A|^{C}. Establish upper and lower bounds for CC that are as strong as possible.

Statement quoted from problem 44 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve improved the best known bound; the record has since been surpassed. The problem remains open. Status in the repository: Former new result, since improved upon. See Section 6.25 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

Notes from the repository

Improved constructions were subsequently obtained in "Sums and differences of sets (improvement over AlphaEvolve)" by Robert Gerbicz and in "Sums and differences of sets: a further improvement over AlphaEvolve" by Fan Zheng. by

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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