Sums and differences of sets (improvement over AlphaEvolve)

TL;DR

This paper presents an improved lower bound for the sum and difference of sets problem, achieving a new bound of 1.173050 using explicit constructions and efficient arithmetic libraries, advancing the state of the art in mathematical bounds.

Contribution

The paper introduces a novel explicit construction that significantly improves the lower bound for the sum and difference of sets problem over previous results.

Findings

New lower bound of 1.173050 achieved

Explicit construction of a large set with over 10^43546 elements

Utilized GMP library for efficient computations

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.