{"paper":{"title":"Conditional expanding bounds for two-variable functions over finite valuation rings","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Le Anh Vinh, Le Quang Ham, Thang Pham","submitted_at":"2015-10-12T22:52:10Z","abstract_excerpt":"In this paper, we use methods from spectral graph theory to obtain some results on the sum-product problem over finite valuation rings $\\mathcal{R}$ of order $q^r$ which generalize recent results given by Hegyv\\'ari and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$, if $A$ and $B$ are two sets in $\\mathcal{R}^*$ with $|A|=|B|=q^\\alpha$, then \\[\\max\\left\\lbrace |f(A, B)|, |g(A, B)| \\right\\rbrace\\gtrsim |A|^{1+\\Delta(\\alpha)},\\] for some $\\Delta(\\alpha)>0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03479","kind":"arxiv","version":2},"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"}