arXiv:2611.05687v1 Announce Type: new Abstract: Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing the archimedean places, and let $\mathcal{O}_{K,S}$ denote the ring of $S$-integers of $K$. We prove that multiplication is positive-existentially definable in $(\mathcal{O}_{K,S},0,1,+,\mid)$ if and only if the unit group $\mathcal{O}_{K,S}^\times$ is infinite. This extends a 1977 theorem of Lipshitz for rings of integers to rings of $S$-integers. One of the key ingredients is a uniform bound for unit equations due to Evertse, Schlickewei and Schmidt, which we use to recover multiplication on the unit group.
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