{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:PZ7ZCNTYB3M2FEYJV3PI2K2KBE","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":"bda40598beb257d88948e97a7092337f46435e71f66ab1e69a41e0ee14d6a7e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-27T17:37:50Z","title_canon_sha256":"a2140f15fffd783ed12f93ae9cfd8042fee71d752b37ce465657ccd6f8c5f301"},"schema_version":"1.0","source":{"id":"1110.6144","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6144","created_at":"2026-05-18T04:10:09Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6144v1","created_at":"2026-05-18T04:10:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6144","created_at":"2026-05-18T04:10:09Z"},{"alias_kind":"pith_short_12","alias_value":"PZ7ZCNTYB3M2","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PZ7ZCNTYB3M2FEYJ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PZ7ZCNTY","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:25f8ea84f1606690a0dfe2be878154d4c6e516f6630ca90a1bd13b4897db3da6","target":"graph","created_at":"2026-05-18T04:10:09Z","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 $P\\subseteq \\mathbb{N}$ and let $(\\Sigma_P,\\,\\sigma_P)$ be the space of a spacing shift. We show that if entropy $h_{\\sigma_P}=0$ then $(\\Sigma_P,\\,\\sigma_P)$ is proximal. Also $h_{\\sigma_P}=0$ if and only if $P=\\mathbb N\\setminus E$ where $E$ is an intersective set. Moreover, we show that\n  $h_{\\sigma_P}>0$ implies that $P$ is a $\\Delta^*$ set; and by giving a class of examples, we show that this is not a sufficient condition. Then there is enough results to solve question 5 given in [J. Banks et al., \\textit{Dynamics of Spacing Shifts}, Discrete Contin. Dyn. Syst., to appear.].","authors_text":"Dawoud Ahmadi Dastjerdi, Maliheh Dabbaghian Amiri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-27T17:37:50Z","title":"Characterization of Entropy for Spacing shifts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6144","kind":"arxiv","version":1},"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":"sha256:6f4aa1e25bed1338bc9903effdd401eb10c64c06458fecba9d4272b0f5d7eaf9","target":"record","created_at":"2026-05-18T04:10:09Z","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":"bda40598beb257d88948e97a7092337f46435e71f66ab1e69a41e0ee14d6a7e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-27T17:37:50Z","title_canon_sha256":"a2140f15fffd783ed12f93ae9cfd8042fee71d752b37ce465657ccd6f8c5f301"},"schema_version":"1.0","source":{"id":"1110.6144","kind":"arxiv","version":1}},"canonical_sha256":"7e7f9136780ed9a29309aede8d2b4a093316dc40f279c0c1a19df34f60b4cb29","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7e7f9136780ed9a29309aede8d2b4a093316dc40f279c0c1a19df34f60b4cb29","first_computed_at":"2026-05-18T04:10:09.467854Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:09.467854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HWusrGMLNW5qP+UG5j//LJTotMGMjK3YFoOA8nJK7dQ58Lq2SEh+B4fDymuR900MVW14tOayLx3bQcbEVzSPBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:09.468536Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.6144","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6f4aa1e25bed1338bc9903effdd401eb10c64c06458fecba9d4272b0f5d7eaf9","sha256:25f8ea84f1606690a0dfe2be878154d4c6e516f6630ca90a1bd13b4897db3da6"],"state_sha256":"4d497e34f6ecfeb594d75327d30891f709a59e4b289ac778ef701a9c2126b5a4"}