{"paper":{"title":"The average number of integral points in orbits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Wade Hindes","submitted_at":"2015-09-02T15:48:31Z","abstract_excerpt":"Over a number field $K$, a celebrated result of Silverman states that if $\\varphi(z)\\in K(z)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit $\\text{Orb}_\\varphi(P)=\\{\\varphi^n(P)\\}_{n\\geq0}$ is finite for all $P\\in \\mathbb{P}^1(K)$. In this paper, we show that if we vary $\\varphi$ and $P$ in a suitable family, the number of $S$-integral points in $\\text{Orb}_\\varphi(P)$ is absolutely bounded. In particular, if we fix $\\varphi$ and vary the basepoint $P\\in \\mathbb{P}^1(K)$, then we show that $\\#(\\text{Orb}_\\varphi(P)\\cap\\mathcal{O}_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00752","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}