{"paper":{"title":"Counting Spectral Radii of Matrices with Positive Entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J. A. Dias da Silva, Pedro J. Freitas","submitted_at":"2013-05-06T10:26:40Z","abstract_excerpt":"The sum-product conjecture of Erd\\H os and Szemer\\'edi states that, given a finite set $A$ of positive numbers, one can find asymptotic lower bounds for $\\max\\{|A+A|,|A\\cdot A|\\}$ of the order of $|A|^{1+\\delta}$ for every $\\delta <1$. In this paper we consider the set of all spectral radii of $n\\times n$ matrices with entries in $A$, and find lower bounds for the cardinality of this set. In the case $n=2$, this cardinality is necessarily larger than $\\max\\{|A+A|,|A\\cdot A|\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1139","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":""},"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"}