{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:LASHICZTIFC4EKVSFOAQWJNDYZ","short_pith_number":"pith:LASHICZT","canonical_record":{"source":{"id":"2002.00455","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-02-02T18:47:52Z","cross_cats_sorted":["math.NT","math.PR"],"title_canon_sha256":"89c15a73c269350cb6daa2fadacd54ddbb97b9a511e7fb96c4be018565d07236","abstract_canon_sha256":"23bf8645fda52974def8636e8fe26152f62d14b39bd64e054329c25c19268e3d"},"schema_version":"1.0"},"canonical_sha256":"5824740b334145c22ab22b810b25a3c65bb36332c0cad48745eaf5b774a763ae","source":{"kind":"arxiv","id":"2002.00455","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2002.00455","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"arxiv_version","alias_value":"2002.00455v3","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2002.00455","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"pith_short_12","alias_value":"LASHICZTIFC4","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"pith_short_16","alias_value":"LASHICZTIFC4EKVS","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"pith_short_8","alias_value":"LASHICZT","created_at":"2026-07-05T04:45:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:LASHICZTIFC4EKVSFOAQWJNDYZ","target":"record","payload":{"canonical_record":{"source":{"id":"2002.00455","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-02-02T18:47:52Z","cross_cats_sorted":["math.NT","math.PR"],"title_canon_sha256":"89c15a73c269350cb6daa2fadacd54ddbb97b9a511e7fb96c4be018565d07236","abstract_canon_sha256":"23bf8645fda52974def8636e8fe26152f62d14b39bd64e054329c25c19268e3d"},"schema_version":"1.0"},"canonical_sha256":"5824740b334145c22ab22b810b25a3c65bb36332c0cad48745eaf5b774a763ae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:45:08.946622Z","signature_b64":"Rrtr/2F430xG9+FtAcjxEc6zPYOlLcvdr6TofxfsL74ZC+P1KlI4Nnfma4vJ/EMR9B/SDfPh8xcRmGdKLiPABA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5824740b334145c22ab22b810b25a3c65bb36332c0cad48745eaf5b774a763ae","last_reissued_at":"2026-07-05T04:45:08.946081Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:45:08.946081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2002.00455","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:45:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"++36JTzWT7zALa4MLI/6ZBwQ3REyx2qVCoP54HOtrmUqqId81Y9CpXz/PTw/DsX7gpzlpr5yaqSGu/XZY7s/DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T13:38:39.929976Z"},"content_sha256":"484a159a496ee0ca87550386ca357d63f1cdf3d461795153e96862e3ac00ddca","schema_version":"1.0","event_id":"sha256:484a159a496ee0ca87550386ca357d63f1cdf3d461795153e96862e3ac00ddca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:LASHICZTIFC4EKVSFOAQWJNDYZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Random walks on tori and normal numbers in self similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.PR"],"primary_cat":"math.DS","authors_text":"Arijit Ganguly, Barak Weiss, Yiftach Dayan","submitted_at":"2020-02-02T18:47:52Z","abstract_excerpt":"We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \\subset \\mathbb{R}^d$ is an attractor of a finite iterated function system of $n\\geq 2$ maps of the form $x \\mapsto D^{-r_i} x + t_i \\ (i=1, \\ldots, n)$, where $D$ is an expanding $d\\times d$ integer matrix, and is the same for all the maps, and $r_{i} \\in\\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2002.00455","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2002.00455/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T04:45:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0bcFPt1SrD7tKELxETM/J7gLJ/EqfBXMtarn5T3bUfr/tZY2OvuiZhpD8a2Xk3P76vZTcSjv4gAqZODMJYzjAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T13:38:39.930348Z"},"content_sha256":"13edc168e35e9c95e7c48a6a6c8f8d74c6883f6d0ce1a7b8d997df88398113e5","schema_version":"1.0","event_id":"sha256:13edc168e35e9c95e7c48a6a6c8f8d74c6883f6d0ce1a7b8d997df88398113e5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LASHICZTIFC4EKVSFOAQWJNDYZ/bundle.json","state_url":"https://pith.science/pith/LASHICZTIFC4EKVSFOAQWJNDYZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LASHICZTIFC4EKVSFOAQWJNDYZ/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":"2026-08-19T13:38:39Z","links":{"resolver":"https://pith.science/pith/LASHICZTIFC4EKVSFOAQWJNDYZ","bundle":"https://pith.science/pith/LASHICZTIFC4EKVSFOAQWJNDYZ/bundle.json","state":"https://pith.science/pith/LASHICZTIFC4EKVSFOAQWJNDYZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LASHICZTIFC4EKVSFOAQWJNDYZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:LASHICZTIFC4EKVSFOAQWJNDYZ","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":"23bf8645fda52974def8636e8fe26152f62d14b39bd64e054329c25c19268e3d","cross_cats_sorted":["math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-02-02T18:47:52Z","title_canon_sha256":"89c15a73c269350cb6daa2fadacd54ddbb97b9a511e7fb96c4be018565d07236"},"schema_version":"1.0","source":{"id":"2002.00455","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2002.00455","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"arxiv_version","alias_value":"2002.00455v3","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2002.00455","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"pith_short_12","alias_value":"LASHICZTIFC4","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"pith_short_16","alias_value":"LASHICZTIFC4EKVS","created_at":"2026-07-05T04:45:08Z"},{"alias_kind":"pith_short_8","alias_value":"LASHICZT","created_at":"2026-07-05T04:45:08Z"}],"graph_snapshots":[{"event_id":"sha256:13edc168e35e9c95e7c48a6a6c8f8d74c6883f6d0ce1a7b8d997df88398113e5","target":"graph","created_at":"2026-07-05T04:45:08Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2002.00455/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \\subset \\mathbb{R}^d$ is an attractor of a finite iterated function system of $n\\geq 2$ maps of the form $x \\mapsto D^{-r_i} x + t_i \\ (i=1, \\ldots, n)$, where $D$ is an expanding $d\\times d$ integer matrix, and is the same for all the maps, and $r_{i} \\in\\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every p","authors_text":"Arijit Ganguly, Barak Weiss, Yiftach Dayan","cross_cats":["math.NT","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-02-02T18:47:52Z","title":"Random walks on tori and normal numbers in self similar sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2002.00455","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":"sha256:484a159a496ee0ca87550386ca357d63f1cdf3d461795153e96862e3ac00ddca","target":"record","created_at":"2026-07-05T04:45:08Z","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":"23bf8645fda52974def8636e8fe26152f62d14b39bd64e054329c25c19268e3d","cross_cats_sorted":["math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-02-02T18:47:52Z","title_canon_sha256":"89c15a73c269350cb6daa2fadacd54ddbb97b9a511e7fb96c4be018565d07236"},"schema_version":"1.0","source":{"id":"2002.00455","kind":"arxiv","version":3}},"canonical_sha256":"5824740b334145c22ab22b810b25a3c65bb36332c0cad48745eaf5b774a763ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5824740b334145c22ab22b810b25a3c65bb36332c0cad48745eaf5b774a763ae","first_computed_at":"2026-07-05T04:45:08.946081Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:45:08.946081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rrtr/2F430xG9+FtAcjxEc6zPYOlLcvdr6TofxfsL74ZC+P1KlI4Nnfma4vJ/EMR9B/SDfPh8xcRmGdKLiPABA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:45:08.946622Z","signed_message":"canonical_sha256_bytes"},"source_id":"2002.00455","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:484a159a496ee0ca87550386ca357d63f1cdf3d461795153e96862e3ac00ddca","sha256:13edc168e35e9c95e7c48a6a6c8f8d74c6883f6d0ce1a7b8d997df88398113e5"],"state_sha256":"69548aaa4bef2a606151d2c5d842ca903d8de7a75cb71309642794fb3cb8cc30"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nlh+UBj13N6l/Yz96W/2n1z6BrpZIKsHpW4xzYx5TvpD9lWrzzhc4D3Gs7kRtx1aMFBGWxHeS9qtUyef7yLmBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T13:38:39.932843Z","bundle_sha256":"14e2718493b1e995c30607d160711a728c88f4798a18b603a802237114e5be82"}}