Sums and differences of sets (improvement over AlphaEvolve)
Robert Gerbicz
TL;DR
The paper tackles the sums and differences of sets problem by refining the classical $V(m,L)$ construction through a restrictive-coordinate approach. It introduces the $W(m,L,B)$ set and the associated $U(m,L,B)$ via the injective map $g$, enabling explicit formulas for $|U|$, $|U+U|$, and $|U-U|$ and a computable bound on $\theta$ through $\theta \ge 1 + \frac{\log\left(\frac{|U-U|}{|U+U|}\right)}{\log(2\max(U)+1)}$. Leveraging both floating-point searches and exact integer arithmetic (via GMP), the authors achieve a new lower bound $\theta=1.173050$ using $m=81411$, $L=65536$, $B=5$, with enormous $|U|$ and sums/differences, surpassing the previous AlphaEvolve result of $\theta=1.1584$. The work demonstrates that larger $L$ may yield tiny gains and provides detailed computational methods and explicit numbers for replication. Overall, it advances the state of the art in explicit constructions for sumset/difference-set growth rates and highlights practical finite-precision considerations in such combinatorial constructions.
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 $θ=1.14465$ to $θ=1.1584$. In this paper, we present an improved bound $θ=1.173050$ 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.