{"paper":{"title":"A Romanoff-type theorem for $P_2$+{$a^a$: a$\\ge$ 1}","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Huixi Li, Junfeng Li, Yuchen Ding","submitted_at":"2026-07-04T02:31:00Z","abstract_excerpt":"Let $\\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\\ge$ 1:$\\Omega(m)$ $\\le$ 2}. We prove that the sumset $P_2$+{$a^a$: a$\\ge$ 1} has positive lower density. The proof uses the Romanoff second moment method, in the spirit of Li and Pan's theorem on $P_2$+$2^{\\mathcal P}$. The main new ingredient is the following average estimate for the singular factor \\[\n  \\frac{1}{K(K-1)}\n  \\sum_{\\substack{1\\le a,b\\le K\\\\a\\ne b}}\n  \\prod_{p\\mid a^a-b^b}\\left(1+\\frac{\\kappa}{p}\\right)\n  \\le C_\\kappa \\] for some constant $C_\\kappa>0$, which is valid for all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03662","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/2607.03662/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"}