On the Extended 1-2-3 Conjecture of Pilz

Philippa Holdridge; Péter Pál Pach

On the Extended 1-2-3 Conjecture of Pilz

Combinatorics 2026-07-01 v1 Number Theory

Authors: Philippa Holdridge , Péter Pál Pach

Abstract

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 $n$ .

Keywords

symmetric difference sumset additive combinatorics Pilz conjecture finite sets of integers

Cite

@article{arxiv.2607.00934,
  title  = {On the Extended 1-2-3 Conjecture of Pilz},
  author = {Philippa Holdridge and Péter Pál Pach},
  journal= {arXiv preprint arXiv:2607.00934},
  year   = {2026}
}

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30 pages