{"paper":{"title":"Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ray Li, Steven J. Miller","submitted_at":"2016-06-27T02:00:17Z","abstract_excerpt":"Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\\{1,2,3,5,\\dots\\}$. This has been extended to many other recurrence relations $\\{G_n\\}$ (with their own notion of a legal decomposition) and to proving that the distribution of the number of summands of an $M \\in [G_n, G_{n+1})$ converges to a Gaussian as $n\\to\\infty$. We prove that for any non-negative integer $g$ the average number of gaps of size $g$ in many generalized Zeckendorf decompositions is $C_\\mu n+d_\\mu+o(1)$ for constants $C_\\mu > 0$ and $d_\\mu$ depending on $g$ and the recurr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.08110","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"}