On the Extended 1-2-3 Conjecture of Pilz

We resolve (for all sufficiently large $n$) a conjecture of Pilz on the symmetric difference $A\mathrm{\Delta }(2A)\mathrm{\Delta }\cdots \mathrm{\Delta }(nA)$ for finite sets $A\subseteq \mathbb{N}$ of positive integers. We show that this set always has cardinality at least $n$ for large $n$.