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In this paper, using exponential sums, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\\hat{r}_{k_1,k_2}(A,n)=\\hat{r}_{k_1,k_2}(\\mathbb{Z}_m\\setminus A,n)$ for all $n\\in \\mathbb{Z}_m$. We also pose several problems for further research."},"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":"1208.4195","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-08-21T05:41:27Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"36d95f208f7ca3ddf2c875eb4bb9409ab06a67022c5e15fcc09dd2077ef8b3a3","abstract_canon_sha256":"4ac0fe6faead4e4568a019312179856074de8853c374a9f552cad2c45abc94d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:54.256863Z","signature_b64":"3z4s1BpqvVjWeM34EpgJJe5Ud+CkslV1fyY8ewxSbcSfvrwmmkTDee/LFVcUZGRS5hT/7vCNcZIcZga4BPjBAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff6438c263fd1e6599057ab7a4879d7ddfe43bbad60f6669970864116ecb955a","last_reissued_at":"2026-05-18T02:42:54.256503Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:54.256503Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted representation functions on $\\mathbb{Z}_m$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Quan-Hui Yang, Yong-Gao Chen","submitted_at":"2012-08-21T05:41:27Z","abstract_excerpt":"Let $m$, $k_1$, and $k_2$ be three integers with $m\\ge 2$. For any set $A\\subseteq \\mathbb{Z}_m$ and $n\\in \\mathbb{Z}_m$, let $\\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\\in A$. In this paper, using exponential sums, we characterize all $m$, $k_1$, $k_2$, and $A$ for which $\\hat{r}_{k_1,k_2}(A,n)=\\hat{r}_{k_1,k_2}(\\mathbb{Z}_m\\setminus A,n)$ for all $n\\in \\mathbb{Z}_m$. 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