{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TALIFYE7OQ2ELEMJZ2ZRPJK7MJ","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":"3a32424b93c1324c82ec33336439b1e7a24102b2b117d6801f17d0740845d86d","cross_cats_sorted":["math.MP","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-07-28T03:01:33Z","title_canon_sha256":"fc7cbd75afc36c676793b185c37f0c6702219fb8b532fa87285e9edc9e45a832"},"schema_version":"1.0","source":{"id":"1207.6681","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6681","created_at":"2026-05-18T03:34:48Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6681v2","created_at":"2026-05-18T03:34:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6681","created_at":"2026-05-18T03:34:48Z"},{"alias_kind":"pith_short_12","alias_value":"TALIFYE7OQ2E","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TALIFYE7OQ2ELEMJ","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TALIFYE7","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:fdffc3c04e018e976f34caecdd336fd0a9057c4a89a5737ec657af26bed7611f","target":"graph","created_at":"2026-05-18T03:34:48Z","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":"We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions; the similarity dimension via the Moran equation (at least in the case of self-similar sets); the order of the (box-)counting function; the classic result on compact subsets of the real line due to Besicovitch and Taylor, as adapted to the theory of fractal strings; and the abscissae of convergence of new classes of zeta functions. Specifically, we define bo","authors_text":"Darko \\v{Z}ubrini\\'c, John A. Rock, Michel L. Lapidus","cross_cats":["math.MP","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-07-28T03:01:33Z","title":"Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6681","kind":"arxiv","version":2},"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:cb985b6c186a272c328e8d6a5af0494c5a35c54fbcd99349494165abdfdb1749","target":"record","created_at":"2026-05-18T03:34:48Z","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":"3a32424b93c1324c82ec33336439b1e7a24102b2b117d6801f17d0740845d86d","cross_cats_sorted":["math.MP","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-07-28T03:01:33Z","title_canon_sha256":"fc7cbd75afc36c676793b185c37f0c6702219fb8b532fa87285e9edc9e45a832"},"schema_version":"1.0","source":{"id":"1207.6681","kind":"arxiv","version":2}},"canonical_sha256":"981682e09f7434459189ceb317a55f62432920fd8afe77ba958886eea607ce53","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"981682e09f7434459189ceb317a55f62432920fd8afe77ba958886eea607ce53","first_computed_at":"2026-05-18T03:34:48.267747Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:48.267747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W8myaJIM0G9Rlutr8SnwEHne19DMTQUg0llFbesG4bj1CJSpIyVNYskbBSepXVNHMvZtoaeNul+LcrIPfKO+Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:48.268383Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.6681","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb985b6c186a272c328e8d6a5af0494c5a35c54fbcd99349494165abdfdb1749","sha256:fdffc3c04e018e976f34caecdd336fd0a9057c4a89a5737ec657af26bed7611f"],"state_sha256":"5509b0157c352ae391f0380f6e60c2ca09d0d3d276b6c7f35caa60837d29e758"}