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In this paper, we prove that, if $A\\subseteq \\mathbb{N}$ and $N$ is a positive integer such that $R_2(A,n)=R_2(\\mathbb{N}\\setminus A,n)$ for all $n\\geq2N-1$, then for any $\\theta$ with $0<\\theta<\\frac{2\\log2-\\log3}{42\\log 2-9\\log3}$, the set of integers $n$ with $R_2(A,n)=\\frac{n}{8}+O(n^{1-\\theta})$ has density one. The similar result holds for $R_3(A,n)$. These improve the results of the first author."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1904.10352","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-04-23T14:23:53Z","cross_cats_sorted":[],"title_canon_sha256":"364ded02b834bd5b4fcb85a91a1e419b1df6fd5521e1d3417ce35e04c53c5fe1","abstract_canon_sha256":"5115578c20391f2028fc3d3fb18081b47567db451db84525ec23837fbd5fa9f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:54.334753Z","signature_b64":"+0oOfjey8aNG8M3RtS5PvchFEABvSZmQyQpsb+6z/w0mJMkKqsfOvt5XH5G7uLyJV78FzxaGCVDIFY6K/bX6Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a749060e1da283820d8e0b717731d6642f5ac8b9591a7f32818e491a9c8ab96","last_reissued_at":"2026-05-17T23:47:54.334260Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:54.334260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the values of representation functions II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Csaba Sandor, Quan-Hui Yang, Xing-Wang Jiang","submitted_at":"2019-04-23T14:23:53Z","abstract_excerpt":"For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\\in A$, $a<a'$ and $a\\leq a'$, respectively. In this paper, we prove that, if $A\\subseteq \\mathbb{N}$ and $N$ is a positive integer such that $R_2(A,n)=R_2(\\mathbb{N}\\setminus A,n)$ for all $n\\geq2N-1$, then for any $\\theta$ with $0<\\theta<\\frac{2\\log2-\\log3}{42\\log 2-9\\log3}$, the set of integers $n$ with $R_2(A,n)=\\frac{n}{8}+O(n^{1-\\theta})$ has density one. The similar result holds for $R_3(A,n)$. 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