{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:3LPNJZHAXG5JHVJDH7DFPNETWI","short_pith_number":"pith:3LPNJZHA","schema_version":"1.0","canonical_sha256":"daded4e4e0b9ba93d5233fc657b493b22bfcd1ba11749bf130449d07a95d91db","source":{"kind":"arxiv","id":"1808.09724","version":1},"attestation_state":"computed","paper":{"title":"Arithmetic representations of real numbers in terms of self-similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Kan Jiang, Lifeng Xi","submitted_at":"2018-08-29T11:02:18Z","abstract_excerpt":"Suppose $n\\geq 2$ and $\\mathcal{A}_{i}\\subset \\{0,1,\\cdots ,(n-1)\\}$ for $ i=1,\\cdots ,l,$ let $K_{i}=\\bigcup\\nolimits_{a\\in \\mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\\cdots ,m_{l}\\in \\mathbb{Z}$ with $\\prod\\nolimits_{i}m_{i}\\neq 0,$ we let \\begin{equation*} S_{x}=\\left\\{ \\mathbf{(}y_{1},\\cdots ,y_{l}\\mathbf{)}:m_{1}y_{1}+\\cdots +m_{l}y_{l}=x\\text{ with }y_{i}\\in K_{i}\\text{ }\\forall i\\right\\} . \\end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set \\begin{equation*} U_{r}=\\{x:\\mathbf{Card}(S_"},"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":"1808.09724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-08-29T11:02:18Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"f97e5add9f6c278d87bdbeaf62b3e8fa1124581c7baba4a67071224df83965b5","abstract_canon_sha256":"26b359dc8a3ebf063157c172d0f4e8ef8ecbb51c64d44304d81da5088c6a3b72"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:54.647105Z","signature_b64":"MrErgkwtNT8OVMQr9Jhr5cyKp161uX+EcyrWcGnNZCzKB0Dbt6Dlsu2iQonbJDHKlAzcpR7n7b5epkpawUzABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"daded4e4e0b9ba93d5233fc657b493b22bfcd1ba11749bf130449d07a95d91db","last_reissued_at":"2026-05-18T00:06:54.646361Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:54.646361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic representations of real numbers in terms of self-similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Kan Jiang, Lifeng Xi","submitted_at":"2018-08-29T11:02:18Z","abstract_excerpt":"Suppose $n\\geq 2$ and $\\mathcal{A}_{i}\\subset \\{0,1,\\cdots ,(n-1)\\}$ for $ i=1,\\cdots ,l,$ let $K_{i}=\\bigcup\\nolimits_{a\\in \\mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\\cdots ,m_{l}\\in \\mathbb{Z}$ with $\\prod\\nolimits_{i}m_{i}\\neq 0,$ we let \\begin{equation*} S_{x}=\\left\\{ \\mathbf{(}y_{1},\\cdots ,y_{l}\\mathbf{)}:m_{1}y_{1}+\\cdots +m_{l}y_{l}=x\\text{ with }y_{i}\\in K_{i}\\text{ }\\forall i\\right\\} . \\end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set \\begin{equation*} 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