{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:WUVHSCTF2LK3XSY4VFCSBSDMJ5","short_pith_number":"pith:WUVHSCTF","canonical_record":{"source":{"id":"1509.04785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-16T01:26:07Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"55205a62e8b2e0c2a63a846f3c53a09f1589f34af631ffee30febee0dbf8c33f","abstract_canon_sha256":"a2d6163ec770a40f2256f58a4daa9f7d57263fb1dd1dd839f5938d4d9341e38d"},"schema_version":"1.0"},"canonical_sha256":"b52a790a65d2d5bbcb1ca94520c86c4f5ebf3d1d5aaffb46a07b03606ec0f81e","source":{"kind":"arxiv","id":"1509.04785","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.04785","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"arxiv_version","alias_value":"1509.04785v2","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.04785","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"pith_short_12","alias_value":"WUVHSCTF2LK3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WUVHSCTF2LK3XSY4","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WUVHSCTF","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:WUVHSCTF2LK3XSY4VFCSBSDMJ5","target":"record","payload":{"canonical_record":{"source":{"id":"1509.04785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-16T01:26:07Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"55205a62e8b2e0c2a63a846f3c53a09f1589f34af631ffee30febee0dbf8c33f","abstract_canon_sha256":"a2d6163ec770a40f2256f58a4daa9f7d57263fb1dd1dd839f5938d4d9341e38d"},"schema_version":"1.0"},"canonical_sha256":"b52a790a65d2d5bbcb1ca94520c86c4f5ebf3d1d5aaffb46a07b03606ec0f81e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:31.601597Z","signature_b64":"BL1B5f+h561SrC9q7Z4UKo3Yo+caduWbTCr2LQsfGqztIyafzceSbSJOcUlaktxYZ7XZxBiXoV+4OjpN4sJ4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b52a790a65d2d5bbcb1ca94520c86c4f5ebf3d1d5aaffb46a07b03606ec0f81e","last_reissued_at":"2026-05-18T00:56:31.600588Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:31.600588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.04785","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":"2026-05-18T00:56:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SSvT5rtReDfRJUOtN1uSW/omR9BgzvI3R3yVNG59lgncbxdVX4vzgDacXCDyLd8t+I5oM9TYR3SztEZWW9u7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:50:30.982727Z"},"content_sha256":"4f7db52296d4ef49c9e8582f0fdfc5983f85f51875029f3e64cd43486515674d","schema_version":"1.0","event_id":"sha256:4f7db52296d4ef49c9e8582f0fdfc5983f85f51875029f3e64cd43486515674d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:WUVHSCTF2LK3XSY4VFCSBSDMJ5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant measure of rotational beta expansion and a problem of Tarski","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Jonathan Caalim, Shigeki Akiyama","submitted_at":"2015-09-16T01:26:07Z","abstract_excerpt":"We study invariant measures of a piecewise expanding map in $\\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it has an absolutely continuous invariant measure equivalent to the $m$-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case $m=2$, we obtain an alternative proof of the result in Akiyama-Caalim:2015 together with some improvement."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04785","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":"2026-05-18T00:56:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VJnLWjjSoGyGl74INM4U+mlLhYr0ODVInIiIOGTZ4XRMQW+aDPzP3MOC6JwKPd3ZhgA8wyQ7oL6QCT9s+XYlBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T23:50:30.983079Z"},"content_sha256":"7820639f4b438f62aa7b30eaa8a1745ce1ac9245b7aaa65f81b588b50accfadd","schema_version":"1.0","event_id":"sha256:7820639f4b438f62aa7b30eaa8a1745ce1ac9245b7aaa65f81b588b50accfadd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WUVHSCTF2LK3XSY4VFCSBSDMJ5/bundle.json","state_url":"https://pith.science/pith/WUVHSCTF2LK3XSY4VFCSBSDMJ5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WUVHSCTF2LK3XSY4VFCSBSDMJ5/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":"2026-06-24T23:50:30Z","links":{"resolver":"https://pith.science/pith/WUVHSCTF2LK3XSY4VFCSBSDMJ5","bundle":"https://pith.science/pith/WUVHSCTF2LK3XSY4VFCSBSDMJ5/bundle.json","state":"https://pith.science/pith/WUVHSCTF2LK3XSY4VFCSBSDMJ5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WUVHSCTF2LK3XSY4VFCSBSDMJ5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WUVHSCTF2LK3XSY4VFCSBSDMJ5","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":"a2d6163ec770a40f2256f58a4daa9f7d57263fb1dd1dd839f5938d4d9341e38d","cross_cats_sorted":["math.MG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-16T01:26:07Z","title_canon_sha256":"55205a62e8b2e0c2a63a846f3c53a09f1589f34af631ffee30febee0dbf8c33f"},"schema_version":"1.0","source":{"id":"1509.04785","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.04785","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"arxiv_version","alias_value":"1509.04785v2","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.04785","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"pith_short_12","alias_value":"WUVHSCTF2LK3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WUVHSCTF2LK3XSY4","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WUVHSCTF","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:7820639f4b438f62aa7b30eaa8a1745ce1ac9245b7aaa65f81b588b50accfadd","target":"graph","created_at":"2026-05-18T00:56:31Z","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 study invariant measures of a piecewise expanding map in $\\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it has an absolutely continuous invariant measure equivalent to the $m$-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case $m=2$, we obtain an alternative proof of the result in Akiyama-Caalim:2015 together with some improvement.","authors_text":"Jonathan Caalim, Shigeki Akiyama","cross_cats":["math.MG","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-16T01:26:07Z","title":"Invariant measure of rotational beta expansion and a problem of Tarski"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04785","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":"sha256:4f7db52296d4ef49c9e8582f0fdfc5983f85f51875029f3e64cd43486515674d","target":"record","created_at":"2026-05-18T00:56:31Z","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":"a2d6163ec770a40f2256f58a4daa9f7d57263fb1dd1dd839f5938d4d9341e38d","cross_cats_sorted":["math.MG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-16T01:26:07Z","title_canon_sha256":"55205a62e8b2e0c2a63a846f3c53a09f1589f34af631ffee30febee0dbf8c33f"},"schema_version":"1.0","source":{"id":"1509.04785","kind":"arxiv","version":2}},"canonical_sha256":"b52a790a65d2d5bbcb1ca94520c86c4f5ebf3d1d5aaffb46a07b03606ec0f81e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b52a790a65d2d5bbcb1ca94520c86c4f5ebf3d1d5aaffb46a07b03606ec0f81e","first_computed_at":"2026-05-18T00:56:31.600588Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:31.600588Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BL1B5f+h561SrC9q7Z4UKo3Yo+caduWbTCr2LQsfGqztIyafzceSbSJOcUlaktxYZ7XZxBiXoV+4OjpN4sJ4Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:31.601597Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.04785","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f7db52296d4ef49c9e8582f0fdfc5983f85f51875029f3e64cd43486515674d","sha256:7820639f4b438f62aa7b30eaa8a1745ce1ac9245b7aaa65f81b588b50accfadd"],"state_sha256":"6efca382386ebbdc07f00dc1b380ac3231cd596f5b17df787735d4c3909dc9de"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B6GfkXv+SAee+gYl2xs8zNeAYRxBomDwnwQ/WE/RwIa/lr/b71AhiRCQg3bdOig2AtGrhGeY5PdxKRB1dkBMCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T23:50:30.984992Z","bundle_sha256":"11d43336f2cc2301bd19bc804b8c053f800dfc13ef1be478f4d76d7b6806a9ab"}}