{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:IRAUOVRUIHYR7R2POT7RXYXYFP","short_pith_number":"pith:IRAUOVRU","canonical_record":{"source":{"id":"2606.08865","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-07T22:33:28Z","cross_cats_sorted":[],"title_canon_sha256":"1c5c780b8e1bbec29f0142d478b35db26883b879709c4a3bd10b798bddf0ef78","abstract_canon_sha256":"1eac11bd08b5488905f0dec3016660c13c742afbe48bcd2e063098dfb2be1237"},"schema_version":"1.0"},"canonical_sha256":"444147563441f11fc74f74ff1be2f82bc16e32e076dad286889d51ddae31acbd","source":{"kind":"arxiv","id":"2606.08865","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08865","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08865v1","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08865","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"pith_short_12","alias_value":"IRAUOVRUIHYR","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"pith_short_16","alias_value":"IRAUOVRUIHYR7R2P","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"pith_short_8","alias_value":"IRAUOVRU","created_at":"2026-06-09T02:07:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:IRAUOVRUIHYR7R2POT7RXYXYFP","target":"record","payload":{"canonical_record":{"source":{"id":"2606.08865","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-07T22:33:28Z","cross_cats_sorted":[],"title_canon_sha256":"1c5c780b8e1bbec29f0142d478b35db26883b879709c4a3bd10b798bddf0ef78","abstract_canon_sha256":"1eac11bd08b5488905f0dec3016660c13c742afbe48bcd2e063098dfb2be1237"},"schema_version":"1.0"},"canonical_sha256":"444147563441f11fc74f74ff1be2f82bc16e32e076dad286889d51ddae31acbd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:43.679222Z","signature_b64":"hLgej1Jf02pNNft12HGK5f6+9kpJvQaD2R5gOde4hAgwa4hUbHiZ2NK3dZLdsM1bkroEe0DzADR5pRwA2EmzBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"444147563441f11fc74f74ff1be2f82bc16e32e076dad286889d51ddae31acbd","last_reissued_at":"2026-06-09T02:07:43.678746Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:43.678746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.08865","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-09T02:07:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Cmt0Li/1MXDgnfs9c2qMQZo1GRfidFj9HOJjlafM9hAQjZ9g316COCCeOtFy5+RtX3wBHRlH0XUfIF3pXj0gDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T15:31:43.162487Z"},"content_sha256":"ae9c6a478dbcce542bba25c574d6057b66b96f9dc7efd1010f08b512637ffc7a","schema_version":"1.0","event_id":"sha256:ae9c6a478dbcce542bba25c574d6057b66b96f9dc7efd1010f08b512637ffc7a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:IRAUOVRUIHYR7R2POT7RXYXYFP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Dirichlet spectrum with respect to $L_1$ norm is $\\left[\\frac12,1\\right]$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nikita Shulga","submitted_at":"2026-06-07T22:33:28Z","abstract_excerpt":"We prove that the one-dimensional Dirichlet spectrum with respect to approximation in $L_1$ norm $\\mathbb{D}^{[1]}$ satisfies $$ \\mathbb{D}^{[1]}=\\left[\\frac12,1\\right]. $$ This is equivalent to the fact that the Minkowski spectrum $\\mathbb M$, associated with the Minkowski diagonal continued fraction, satisfies $$\n  \\mathbb M=\\left[\\frac14,\\frac12\\right]. $$ Further, we show that level sets $$\n  \\Theta_m=\\{\\alpha\\in(0,1)\\setminus\\mathbb Q:\\mathfrak m(\\alpha)=m\\}, $$ where $\\mathfrak{m}(\\alpha)$ is the Minkowski constant of $\\alpha$, have Hausdorff dimension strictly greater than $1/2$ for any"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08865","kind":"arxiv","version":1},"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/2606.08865/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":"2026-06-09T02:07:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WJnt5pyAgh4dh3lNtjjDYGnNJ3vS0f0nIgsJMrirzwZB/I8RzfQiU/TFgZLhqJx9cuUnEOlzhViN877afo7xDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T15:31:43.163493Z"},"content_sha256":"3e7eba0241dd4ccc42a3e3fe99d1729995334cc25b02ea7366e0d1587f83ae45","schema_version":"1.0","event_id":"sha256:3e7eba0241dd4ccc42a3e3fe99d1729995334cc25b02ea7366e0d1587f83ae45"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IRAUOVRUIHYR7R2POT7RXYXYFP/bundle.json","state_url":"https://pith.science/pith/IRAUOVRUIHYR7R2POT7RXYXYFP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IRAUOVRUIHYR7R2POT7RXYXYFP/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":"2026-08-13T15:31:43Z","links":{"resolver":"https://pith.science/pith/IRAUOVRUIHYR7R2POT7RXYXYFP","bundle":"https://pith.science/pith/IRAUOVRUIHYR7R2POT7RXYXYFP/bundle.json","state":"https://pith.science/pith/IRAUOVRUIHYR7R2POT7RXYXYFP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IRAUOVRUIHYR7R2POT7RXYXYFP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:IRAUOVRUIHYR7R2POT7RXYXYFP","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":"1eac11bd08b5488905f0dec3016660c13c742afbe48bcd2e063098dfb2be1237","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-07T22:33:28Z","title_canon_sha256":"1c5c780b8e1bbec29f0142d478b35db26883b879709c4a3bd10b798bddf0ef78"},"schema_version":"1.0","source":{"id":"2606.08865","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08865","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08865v1","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08865","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"pith_short_12","alias_value":"IRAUOVRUIHYR","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"pith_short_16","alias_value":"IRAUOVRUIHYR7R2P","created_at":"2026-06-09T02:07:43Z"},{"alias_kind":"pith_short_8","alias_value":"IRAUOVRU","created_at":"2026-06-09T02:07:43Z"}],"graph_snapshots":[{"event_id":"sha256:3e7eba0241dd4ccc42a3e3fe99d1729995334cc25b02ea7366e0d1587f83ae45","target":"graph","created_at":"2026-06-09T02:07:43Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.08865/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that the one-dimensional Dirichlet spectrum with respect to approximation in $L_1$ norm $\\mathbb{D}^{[1]}$ satisfies $$ \\mathbb{D}^{[1]}=\\left[\\frac12,1\\right]. $$ This is equivalent to the fact that the Minkowski spectrum $\\mathbb M$, associated with the Minkowski diagonal continued fraction, satisfies $$\n  \\mathbb M=\\left[\\frac14,\\frac12\\right]. $$ Further, we show that level sets $$\n  \\Theta_m=\\{\\alpha\\in(0,1)\\setminus\\mathbb Q:\\mathfrak m(\\alpha)=m\\}, $$ where $\\mathfrak{m}(\\alpha)$ is the Minkowski constant of $\\alpha$, have Hausdorff dimension strictly greater than $1/2$ for any","authors_text":"Nikita Shulga","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-07T22:33:28Z","title":"The Dirichlet spectrum with respect to $L_1$ norm is $\\left[\\frac12,1\\right]$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08865","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:ae9c6a478dbcce542bba25c574d6057b66b96f9dc7efd1010f08b512637ffc7a","target":"record","created_at":"2026-06-09T02:07:43Z","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":"1eac11bd08b5488905f0dec3016660c13c742afbe48bcd2e063098dfb2be1237","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-07T22:33:28Z","title_canon_sha256":"1c5c780b8e1bbec29f0142d478b35db26883b879709c4a3bd10b798bddf0ef78"},"schema_version":"1.0","source":{"id":"2606.08865","kind":"arxiv","version":1}},"canonical_sha256":"444147563441f11fc74f74ff1be2f82bc16e32e076dad286889d51ddae31acbd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"444147563441f11fc74f74ff1be2f82bc16e32e076dad286889d51ddae31acbd","first_computed_at":"2026-06-09T02:07:43.678746Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:07:43.678746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hLgej1Jf02pNNft12HGK5f6+9kpJvQaD2R5gOde4hAgwa4hUbHiZ2NK3dZLdsM1bkroEe0DzADR5pRwA2EmzBQ==","signature_status":"signed_v1","signed_at":"2026-06-09T02:07:43.679222Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.08865","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ae9c6a478dbcce542bba25c574d6057b66b96f9dc7efd1010f08b512637ffc7a","sha256:3e7eba0241dd4ccc42a3e3fe99d1729995334cc25b02ea7366e0d1587f83ae45"],"state_sha256":"a1eca380b0001446059695f269a61cd63ed00ce25a7775efb8684bae813d497a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bojdR1YSX4Dd5fC6vivN1CRr9JrUOuqgTNSUsSl+eLBGKL1mI0mqALSYuN2FaDfjkAzTpPibT1lDIt28v5BZDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T15:31:43.170743Z","bundle_sha256":"7dc0d4c97bbeffada6693ffc03b5c18d20b9acd93a9fbebda93fc96ddb14dc27"}}