{"paper":{"title":"Adjacent comparison bounds and extremal sets for Ruzsa numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Huixi Li, Junfeng Li, Wei Niu, Xiamiao Zhao, Yuchen Ding","submitted_at":"2026-06-09T16:31:06Z","abstract_excerpt":"Let $m$ be a positive integer and $\\mathbb{Z}_m$ the residue class ring modulo $m$. The Ruzsa number $R_m$ is defined to be the least integer $r$ such that there is a subset $\\mathcal{A}$ of $\\mathbb{Z}_m$ satisfying $\n1\\le \\sigma_{\\mathcal{A}}(n)\\le r $\nfor any $n\\in \\mathbb{Z}_m$, where $$ \\sigma_{\\mathcal{A}}(n) =\\#\\big\\{(a,a')\\in\\mathcal A^2:\n  a+a'\\equiv n\\pmod{m}\\big\\}. $$ Motivated by a 2024 conjecture of Ding and Zhao, we prove $\n| R_{m+1}-R_m|\\le 144. $ Let $\\mathcal{A}$ be a subset of $\\mathbb{Z}_m$ satisfying $1\\le \\sigma_{\\mathcal{A}}(n)\\le R_m$ for any $n\\in \\mathbb{Z}_m$. We also"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11069","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/2606.11069/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"}