Sums and differences of sets (improvement over AlphaEvolve)

Abstract

On May 14, 2025, DeepMind announced that AlphaEvolve, a large language model applied to a set of mathematical problems, had matched or exceeded the best known bounds on several problems. In the case of the sum and difference of sets problem, AlphaEvolve, using a set of $54265$ integers, improved the known lower bound of $\theta=1.14465$ to $\theta=1.1584$. In this paper, we present an improved bound $\theta=1.173050$ using an explicit construction of a U set that contains more than $10^{43546}$ elements. For fast integer and floating-point arithmetic, we used the (free) GMP library.