2023/03/27 by Benjamin Bedert, Bedert, Benjamin
Mathematics · #Advanced Topology and Set Theory #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2303.15134
Let A be a subset of the cyclic group Z/pZ with p prime. It is a well-studied problem to determine how small |A| can be if there is no unique sum in A+A, meaning that for every two elements a1,a2∈ A, there exist a1',a2'∈ A such that a1+a2=a1'+a2' and \a1,a2\≠ \a1',a2'\. Let m(p) be the size of a smallest subset of Z/pZ with no unique sum. The previous best known bounds are log p ≪ m(p)≪ √(p). In this paper we improve both the upper and lower bounds to ω(p)log p \leqslant m(p)≪ (log p)2 for some function ω(p) which tends to infinity as p→ ∞. In particular, this shows that for any B⊂ Z/pZ of size |B|<ω(p)log p, its sumset B+B contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum.