An AI research agent closes a 50-year-old sumset bound

Researchers using Tencent's research agent Hyra report settling a question about sums and differences of integers. For a finite set A, count its distinct pairwise sums and its distinct pairwise differences, and divide each count by the size of A. A classical inequality bounds the first ratio by the square of the second. The paper constructs sets showing no smaller universal exponent works, so the best exponent is 2. Paper.

The paper says Hyra, powered by the Hy3 model, searched for the construction, and that model-based judgments only guided exploration rather than certifying correctness. The authors report independently checking the construction and proof and preparing the argument manually. The result therefore rests on the explicit proof, with the AI serving as a discovery aid. Account of AI use.