On the Extended 1-2-3 Conjecture of Pilz

We resolve (for all sufficiently large $n$) a conjecture of Pilz on the symmetric difference $A\Delta (2A)\Delta \cdots\Delta (nA)$ for finite sets $A\subseteq \mathbb{N}$ of positive integers.

We show that this set always has cardinality at least $n$ for large $n$.