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In this paper, we prove that for any $\\theta\\in(0,\\pi)$ and $K_1, K_2\\in \\mathcal{A}$, $Proj_{\\theta}(K_1\\times K_2)$ is similar to a self-similar set or an attractor of some infinite iterated function system, where $Proj_{\\theta}$ denotes the orthogonal projection onto $L_{\\theta}$, and $L_{\\theta}$ denotes the line through the origin in direction $"},"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":"1806.01080","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-06-04T12:48:55Z","cross_cats_sorted":[],"title_canon_sha256":"9c802985e3346264e956cdf1374c99add1f772a217bf68b81652e66eb74fb95c","abstract_canon_sha256":"65c8e1bb06daae1c8548fb5110616925c889809082b63bfeecb5b1d8119c38c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:36.673507Z","signature_b64":"5W07XYjI/7dq99U+YFJW5y79PF+Jat6FbBUJgS3VTg17Xv3WnCaHC3yr2XE6PHB05A35LvzGtZGicpdufLOtBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e98523f9b2d41c8640baf358041a64e9c9beb58e287c6a0303988cfb31ecf111","last_reissued_at":"2026-05-18T00:13:36.672869Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:36.672869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Projections of cartesian products of the self-similar sets without the irrationality assumption","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Kan Jiang","submitted_at":"2018-06-04T12:48:55Z","abstract_excerpt":"Let $\\beta>1$. Define a class of similitudes \\[S=\\left\\{f_{i}(x)=\\dfrac{x}{\\beta^{n_i}}+a_i:n_i\\in \\mathbb{N}^{+}, a_i\\in \\mathbb{R}\\right\\}.\\] Let $\\mathcal{A}$ be the collection of all the self-similar sets generated by the similitudes from $S$. In this paper, we prove that for any $\\theta\\in(0,\\pi)$ and $K_1, K_2\\in \\mathcal{A}$, $Proj_{\\theta}(K_1\\times K_2)$ is similar to a self-similar set or an attractor of some infinite iterated function system, where $Proj_{\\theta}$ denotes the orthogonal projection onto $L_{\\theta}$, and $L_{\\theta}$ denotes the line through the origin in direction $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01080","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1806.01080","created_at":"2026-05-18T00:13:36.672987+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.01080v2","created_at":"2026-05-18T00:13:36.672987+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01080","created_at":"2026-05-18T00:13:36.672987+00:00"},{"alias_kind":"pith_short_12","alias_value":"5GCSH6NS2QOI","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"5GCSH6NS2QOIMQF2","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"5GCSH6NS","created_at":"2026-05-18T12:32:08.215937+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H","json":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H.json","graph_json":"https://pith.science/api/pith-number/5GCSH6NS2QOIMQF26NMAIGTE5H/graph.json","events_json":"https://pith.science/api/pith-number/5GCSH6NS2QOIMQF26NMAIGTE5H/events.json","paper":"https://pith.science/paper/5GCSH6NS"},"agent_actions":{"view_html":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H","download_json":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H.json","view_paper":"https://pith.science/paper/5GCSH6NS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.01080&json=true","fetch_graph":"https://pith.science/api/pith-number/5GCSH6NS2QOIMQF26NMAIGTE5H/graph.json","fetch_events":"https://pith.science/api/pith-number/5GCSH6NS2QOIMQF26NMAIGTE5H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H/action/storage_attestation","attest_author":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H/action/author_attestation","sign_citation":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H/action/citation_signature","submit_replication":"https://pith.science/pith/5GCSH6NS2QOIMQF26NMAIGTE5H/action/replication_record"}},"created_at":"2026-05-18T00:13:36.672987+00:00","updated_at":"2026-05-18T00:13:36.672987+00:00"}