{"paper":{"title":"Trace spectra of simplices in large sets","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Alexander Fish, Michael Bj\\\"orklund, Shrey Sanadhya","submitted_at":"2026-06-11T06:20:02Z","abstract_excerpt":"Given an ordered tuple $\\mathbf v=(v_0,\\ldots,v_d)$ of vectors in $\\mathbb{R}^d$, let $A_{\\mathbf v}=[\\,v_1-v_0\\ \\cdots\\ v_d-v_0\\,]$ be its edge matrix. We prove that, in every finite colouring of $\\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \\[\n  (c_2(A_{\\mathbf v}),\\ldots,c_d(A_{\\mathbf v})). \\] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\\mathbf v})=\\det(A_{\\mathbf v})$. We also prove a discrete analogue: if $E\\subseteq\\mathbb{Z}^d$ has positive upper Banach density, then, for some $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12947","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.12947/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}