{"paper":{"title":"Vector-valued smoothing for finite Sidon sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hongbin Zhao, Jianfeng Hou","submitted_at":"2026-07-01T16:59:14Z","abstract_excerpt":"Let $F(N)$ denote the largest cardinality of a Sidon subset of $\\{0, 1, \\dots, N - 1\\}$. We prove \\[\n  F(N) \\le N^{1/2} + 0.94601 N^{1/4} + O(1). \\] This improves the recently announced coefficient $0.97633$ obtained by Carter, Georgiev, G\\'{o}mez-Serrano, Hunter, O'Bryant, Tao and Wagner. It is also very close to, and numerically below, the tentatively reported value of approximately $0.947$. The argument is based on a vector-valued convolution inequality: several smoothing kernels share the task of producing a boundary majorant, while their $L^2$ energies are averaged. The analytic reduction"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01169","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/2607.01169/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"}