{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:4XLXSTGTGAJI7QOLN3K5DAODYS","short_pith_number":"pith:4XLXSTGT","canonical_record":{"source":{"id":"math/0611599","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2006-11-20T09:09:42Z","cross_cats_sorted":[],"title_canon_sha256":"3dd410de41a19a2778e0232bbfb7fc19fc4f48b99e3265bcce36a768cff9cd3c","abstract_canon_sha256":"d1dae3218876e237bcb3b21c316201b8a7325bef1bd668654ca24c116ec9e331"},"schema_version":"1.0"},"canonical_sha256":"e5d7794cd330128fc1cb6ed5d181c3c49fcb707349676db5e32c5b6f4739562e","source":{"kind":"arxiv","id":"math/0611599","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611599","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611599v2","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611599","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"pith_short_12","alias_value":"4XLXSTGTGAJI","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"4XLXSTGTGAJI7QOL","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"4XLXSTGT","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:4XLXSTGTGAJI7QOLN3K5DAODYS","target":"record","payload":{"canonical_record":{"source":{"id":"math/0611599","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2006-11-20T09:09:42Z","cross_cats_sorted":[],"title_canon_sha256":"3dd410de41a19a2778e0232bbfb7fc19fc4f48b99e3265bcce36a768cff9cd3c","abstract_canon_sha256":"d1dae3218876e237bcb3b21c316201b8a7325bef1bd668654ca24c116ec9e331"},"schema_version":"1.0"},"canonical_sha256":"e5d7794cd330128fc1cb6ed5d181c3c49fcb707349676db5e32c5b6f4739562e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:43.951026Z","signature_b64":"LUnlJ6/A4rTWVG3XUqrIci/P7p/FDRYGBG7bcVjocSptfPqVMy7m2WL90B6Qax60zxcyCcvwXS/1T6NiL3P1DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5d7794cd330128fc1cb6ed5d181c3c49fcb707349676db5e32c5b6f4739562e","last_reissued_at":"2026-05-18T03:39:43.950604Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:43.950604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0611599","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:39:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4+F7JET1MaDK2EPt1bkl2H2MXp/6wmGTdMYIZTB3M16WpMqlR4bNhawjZE3hvrmeyIU8s49neN5AfLPC7IcGCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T08:28:03.177620Z"},"content_sha256":"612567097545459219165a72068279fb7ba300ba548efea9c5a0260302c2640e","schema_version":"1.0","event_id":"sha256:612567097545459219165a72068279fb7ba300ba548efea9c5a0260302c2640e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:4XLXSTGTGAJI7QOLN3K5DAODYS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximation and billiards","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"CPT, FRUMAM), Serge Troubetzkoy (IML","submitted_at":"2006-11-20T09:09:42Z","abstract_excerpt":"This survey is based on a series of talks I gave at the conference \"Dynamical systems and diophantine approximation\" at l'Instut Henri Poincar\\'e in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundance of periodic trajectories in certain irrational polygons."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611599","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:39:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tYwA0v0dBRChlnMdbYhSsgAH+X9iUgF810px0qkHVW9PXUylLa6cKfgWglkAHtxPIr/81ge/Y2xiQcKRFFu+Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T08:28:03.177970Z"},"content_sha256":"a5674b2a1910767e301c80265bfa853220d36ec27ea71d7281c93cbbe119cfdf","schema_version":"1.0","event_id":"sha256:a5674b2a1910767e301c80265bfa853220d36ec27ea71d7281c93cbbe119cfdf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4XLXSTGTGAJI7QOLN3K5DAODYS/bundle.json","state_url":"https://pith.science/pith/4XLXSTGTGAJI7QOLN3K5DAODYS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4XLXSTGTGAJI7QOLN3K5DAODYS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T08:28:03Z","links":{"resolver":"https://pith.science/pith/4XLXSTGTGAJI7QOLN3K5DAODYS","bundle":"https://pith.science/pith/4XLXSTGTGAJI7QOLN3K5DAODYS/bundle.json","state":"https://pith.science/pith/4XLXSTGTGAJI7QOLN3K5DAODYS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4XLXSTGTGAJI7QOLN3K5DAODYS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:4XLXSTGTGAJI7QOLN3K5DAODYS","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":"d1dae3218876e237bcb3b21c316201b8a7325bef1bd668654ca24c116ec9e331","cross_cats_sorted":[],"license":"","primary_cat":"math.DS","submitted_at":"2006-11-20T09:09:42Z","title_canon_sha256":"3dd410de41a19a2778e0232bbfb7fc19fc4f48b99e3265bcce36a768cff9cd3c"},"schema_version":"1.0","source":{"id":"math/0611599","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611599","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611599v2","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611599","created_at":"2026-05-18T03:39:43Z"},{"alias_kind":"pith_short_12","alias_value":"4XLXSTGTGAJI","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"4XLXSTGTGAJI7QOL","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"4XLXSTGT","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:a5674b2a1910767e301c80265bfa853220d36ec27ea71d7281c93cbbe119cfdf","target":"graph","created_at":"2026-05-18T03:39:43Z","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":"This survey is based on a series of talks I gave at the conference \"Dynamical systems and diophantine approximation\" at l'Instut Henri Poincar\\'e in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundance of periodic trajectories in certain irrational polygons.","authors_text":"CPT, FRUMAM), Serge Troubetzkoy (IML","cross_cats":[],"headline":"","license":"","primary_cat":"math.DS","submitted_at":"2006-11-20T09:09:42Z","title":"Approximation and billiards"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611599","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":"sha256:612567097545459219165a72068279fb7ba300ba548efea9c5a0260302c2640e","target":"record","created_at":"2026-05-18T03:39:43Z","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":"d1dae3218876e237bcb3b21c316201b8a7325bef1bd668654ca24c116ec9e331","cross_cats_sorted":[],"license":"","primary_cat":"math.DS","submitted_at":"2006-11-20T09:09:42Z","title_canon_sha256":"3dd410de41a19a2778e0232bbfb7fc19fc4f48b99e3265bcce36a768cff9cd3c"},"schema_version":"1.0","source":{"id":"math/0611599","kind":"arxiv","version":2}},"canonical_sha256":"e5d7794cd330128fc1cb6ed5d181c3c49fcb707349676db5e32c5b6f4739562e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5d7794cd330128fc1cb6ed5d181c3c49fcb707349676db5e32c5b6f4739562e","first_computed_at":"2026-05-18T03:39:43.950604Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:43.950604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LUnlJ6/A4rTWVG3XUqrIci/P7p/FDRYGBG7bcVjocSptfPqVMy7m2WL90B6Qax60zxcyCcvwXS/1T6NiL3P1DA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:43.951026Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0611599","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:612567097545459219165a72068279fb7ba300ba548efea9c5a0260302c2640e","sha256:a5674b2a1910767e301c80265bfa853220d36ec27ea71d7281c93cbbe119cfdf"],"state_sha256":"4d14a60ee2068299bab048e0b99db753b46b28bed17e7ca3ed89bca0ceda0f61"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z6lfAYu+q+pvlpS/1NdhMzcSRl8ANf14Lsy3l7ihAgLvKAbb2vkxav2aIg/Ks4wrLdymEl9Qfr6ZYKPTF+PTCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T08:28:03.179945Z","bundle_sha256":"034377b14d2f70937e8f5f652a2172bb6bdc5ce80be2b8255c5c587d2ecac9e4"}}