{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QVEVBWO5LSXI5ONWY62LTLJV5S","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2e584339adb3fc0843e85a1ce782cf7f18b1d5a41462e7f46a4dde8be3b6a861","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-28T09:47:03Z","title_canon_sha256":"9933d758f21a8663fc5a2eda93056d74721acf05bd9e1a9ff2361dcf5b9a6d37"},"schema_version":"1.0","source":{"id":"1504.07396","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.07396","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"arxiv_version","alias_value":"1504.07396v3","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07396","created_at":"2026-05-17T23:54:27Z"},{"alias_kind":"pith_short_12","alias_value":"QVEVBWO5LSXI","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"QVEVBWO5LSXI5ONW","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"QVEVBWO5","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:54e0ac2d8fd8740d3fbe705b766077e08fe531da6463e34e0f4f3bac86de90d4","target":"graph","created_at":"2026-05-17T23:54:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Suppose that the set ${\\mathcal{T}}= \\{T_1, T_2,...,T_q \\} $ of real $n\\times n$ matrices has joint spectral radius less than $1$. Then for any digit set $ D= \\{d_1, \\cdots, d_q\\} \\subset {\\Bbb R}^n$, there exists a unique nonempty compact set $F=F({\\mathcal{T}},D)$ satisfying $ F = \\bigcup _{j =1}^q T_j(F + d_j)$, which is called a self-affine fractal. We consider an existing criterion for the convex hull of $F$ to be a polytope, which is due to Kirat and Kocyigit. In this note, we strengthen our criterion for the case $T_1=T_2=\\cdots =T_q $. More specifically, we give an upper bound for the ","authors_text":"Ibrahim Kirat, Ilker Kocyigit","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-28T09:47:03Z","title":"On the Convex Hulls of Self-Affine Fractals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07396","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d5f645c31c0bbac0b8c013aadd6e15ac7aa1c8a781c952897a16658e88624a00","target":"record","created_at":"2026-05-17T23:54:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2e584339adb3fc0843e85a1ce782cf7f18b1d5a41462e7f46a4dde8be3b6a861","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-28T09:47:03Z","title_canon_sha256":"9933d758f21a8663fc5a2eda93056d74721acf05bd9e1a9ff2361dcf5b9a6d37"},"schema_version":"1.0","source":{"id":"1504.07396","kind":"arxiv","version":3}},"canonical_sha256":"854950d9dd5cae8eb9b6c7b4b9ad35ecad934f582dd9f3ebcad30c02377565a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"854950d9dd5cae8eb9b6c7b4b9ad35ecad934f582dd9f3ebcad30c02377565a9","first_computed_at":"2026-05-17T23:54:27.270069Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:27.270069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DOQwMTdbn8CBD5oN7UnLSi/sXOUXg87BYdrUcPNZKQAkqaVgFxyaW2C5KTvav4CKRyH6QHQfOdbXjWpd9qflBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:27.270752Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.07396","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d5f645c31c0bbac0b8c013aadd6e15ac7aa1c8a781c952897a16658e88624a00","sha256:54e0ac2d8fd8740d3fbe705b766077e08fe531da6463e34e0f4f3bac86de90d4"],"state_sha256":"f1be5d32d6e7320458a55d511e58e4c849f7a65c0050b18e92572f9d8f6c6a9c"}