{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7NR6TPMDPH5Z46MVT2YQCTQ4SF","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":"73512c927d341a0f71a46c4c4c1f06799bf37b2eeac6b75e1b9ed79a276a1e1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-23T06:47:41Z","title_canon_sha256":"dfba813eb161b89d7c04d22edbd0a8e20c3acce44b9b1b74620eb0a3efaca722"},"schema_version":"1.0","source":{"id":"1009.4528","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4528","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4528v1","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4528","created_at":"2026-05-18T04:40:30Z"},{"alias_kind":"pith_short_12","alias_value":"7NR6TPMDPH5Z","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7NR6TPMDPH5Z46MV","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7NR6TPMD","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:b1398a36f9976222f256a41af68090a870173f5ea5b2e4505ff07e431f9cd582","target":"graph","created_at":"2026-05-18T04:40:30Z","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 prove that there exist arbitrarily small positive real numbers $\\epsilon$ such that every integral power $(1 + \\vepsilon)^n$ is at a distance greater than $2^{-17} \\epsilon |\\log \\vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\\log \\epsilon|^{-1}$. We also establish that the set of real numbers $\\alpha > 1$ such that the sequence of fractional parts $(\\{\\alpha^n\\})_{n \\ge 1}$ is not dense modulo 1 has full Hausdorff dimension.","authors_text":"Nikolay Moshchevitin, Yann Bugeaud","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-23T06:47:41Z","title":"On fractional parts of powers of real numbers close to 1"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4528","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":"sha256:93d0ce6972eb37a71ed888c244d57d93fbda8013ee89db665b1322d7a461bdc1","target":"record","created_at":"2026-05-18T04:40:30Z","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":"73512c927d341a0f71a46c4c4c1f06799bf37b2eeac6b75e1b9ed79a276a1e1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-09-23T06:47:41Z","title_canon_sha256":"dfba813eb161b89d7c04d22edbd0a8e20c3acce44b9b1b74620eb0a3efaca722"},"schema_version":"1.0","source":{"id":"1009.4528","kind":"arxiv","version":1}},"canonical_sha256":"fb63e9bd8379fb9e79959eb1014e1c9161ca8a4a9a671299f5f284cdbd7ed731","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb63e9bd8379fb9e79959eb1014e1c9161ca8a4a9a671299f5f284cdbd7ed731","first_computed_at":"2026-05-18T04:40:30.224962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:30.224962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4bTOyWDzvx/jdMzVIPWELShf5RavHwysMj+/IPHn1wBLnHZOACgpMbiBocW2GzCMdtWDXpXb7++duthLLYvlCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:30.225593Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.4528","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93d0ce6972eb37a71ed888c244d57d93fbda8013ee89db665b1322d7a461bdc1","sha256:b1398a36f9976222f256a41af68090a870173f5ea5b2e4505ff07e431f9cd582"],"state_sha256":"5c81f4cf6960b707ef0f8dcc90e2cb76b02b1a14e324c48136021f392b324c65"}