{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ATABXIG63F2IWPEJ5L5KOTQUQT","short_pith_number":"pith:ATABXIG6","canonical_record":{"source":{"id":"2606.07028","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-05T08:18:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"800fafcc82d8ca43a949abd1ceeed4c19fe131f43055544b72c4d78a0c59f9a9","abstract_canon_sha256":"a15db85fa20b3f53acf7ae603ec635c2e580c80012bbc99c46bcd7a2964f8685"},"schema_version":"1.0"},"canonical_sha256":"04c01ba0ded9748b3c89eafaa74e1484edd0b8843c28f21fc3d0296afdd45328","source":{"kind":"arxiv","id":"2606.07028","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07028","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07028v1","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07028","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"pith_short_12","alias_value":"ATABXIG63F2I","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"pith_short_16","alias_value":"ATABXIG63F2IWPEJ","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"pith_short_8","alias_value":"ATABXIG6","created_at":"2026-06-08T01:04:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ATABXIG63F2IWPEJ5L5KOTQUQT","target":"record","payload":{"canonical_record":{"source":{"id":"2606.07028","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-05T08:18:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"800fafcc82d8ca43a949abd1ceeed4c19fe131f43055544b72c4d78a0c59f9a9","abstract_canon_sha256":"a15db85fa20b3f53acf7ae603ec635c2e580c80012bbc99c46bcd7a2964f8685"},"schema_version":"1.0"},"canonical_sha256":"04c01ba0ded9748b3c89eafaa74e1484edd0b8843c28f21fc3d0296afdd45328","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:42.128524Z","signature_b64":"+BlcxfEdaLGU+8TdwthUqV6SdjonHHj18mJ/3PKWqD3GsiymzHnNLlH8MST4LjB/4yABP+B/7FCBLbUcXvCFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04c01ba0ded9748b3c89eafaa74e1484edd0b8843c28f21fc3d0296afdd45328","last_reissued_at":"2026-06-08T01:04:42.127582Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:42.127582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.07028","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":"2026-06-08T01:04:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HKTBq+RYBl+uoVzNAffkqI3rkUi5VJ9bSvdPcRhbucp8pj7DTI96rLscFdOrfkc59wXBFGt/Vgi4RBRoQLJoDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T05:34:32.394624Z"},"content_sha256":"7aaf49450996902cd47efd7bfed937b4b28511def95c938bf405118224ec0850","schema_version":"1.0","event_id":"sha256:7aaf49450996902cd47efd7bfed937b4b28511def95c938bf405118224ec0850"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ATABXIG63F2IWPEJ5L5KOTQUQT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrepancy estimates for multi-dimensional non-smooth convex bodies: a case study","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Alessandro Monguzzi, Luca Brandolini, Roberto Bramati","submitted_at":"2026-06-05T08:18:37Z","abstract_excerpt":"We study $L^2$-averaged discrepancies of finite sequences of points in the torus $\\mathbb{T}^d$ with respect to translated and dilated copies of convex bodies with non-smooth boundary. Under suitable anisotropic assumptions on the decay of the Fourier transform of the body, we prove matching lower and upper bounds for the averaged discrepancy, obtaining the rate $ N^{1 - \\frac{d+1}{d^2+d-1}}$. This yields an intermediate regime between smooth convex bodies and polytopes and recovers the known exponent $2/5$ in dimension $d=2$. The argument relies on harmonic analysis techniques combined with a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07028","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/2606.07028/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":"2026-06-08T01:04:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uTDHyheTiV7ESh4nnFXaj/o5QZrGzk688GByytG/3XWbZrfklltrHQKfnNbTEVGJZlokXFF1FO0q4RqisAwMCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T05:34:32.395003Z"},"content_sha256":"0945ef65caa0beeb6994e4053a0d4822952b3bb707fc954fd3f2ce43369e266b","schema_version":"1.0","event_id":"sha256:0945ef65caa0beeb6994e4053a0d4822952b3bb707fc954fd3f2ce43369e266b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ATABXIG63F2IWPEJ5L5KOTQUQT/bundle.json","state_url":"https://pith.science/pith/ATABXIG63F2IWPEJ5L5KOTQUQT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ATABXIG63F2IWPEJ5L5KOTQUQT/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":"2026-06-21T05:34:32Z","links":{"resolver":"https://pith.science/pith/ATABXIG63F2IWPEJ5L5KOTQUQT","bundle":"https://pith.science/pith/ATABXIG63F2IWPEJ5L5KOTQUQT/bundle.json","state":"https://pith.science/pith/ATABXIG63F2IWPEJ5L5KOTQUQT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ATABXIG63F2IWPEJ5L5KOTQUQT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ATABXIG63F2IWPEJ5L5KOTQUQT","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":"a15db85fa20b3f53acf7ae603ec635c2e580c80012bbc99c46bcd7a2964f8685","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-05T08:18:37Z","title_canon_sha256":"800fafcc82d8ca43a949abd1ceeed4c19fe131f43055544b72c4d78a0c59f9a9"},"schema_version":"1.0","source":{"id":"2606.07028","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07028","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07028v1","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07028","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"pith_short_12","alias_value":"ATABXIG63F2I","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"pith_short_16","alias_value":"ATABXIG63F2IWPEJ","created_at":"2026-06-08T01:04:42Z"},{"alias_kind":"pith_short_8","alias_value":"ATABXIG6","created_at":"2026-06-08T01:04:42Z"}],"graph_snapshots":[{"event_id":"sha256:0945ef65caa0beeb6994e4053a0d4822952b3bb707fc954fd3f2ce43369e266b","target":"graph","created_at":"2026-06-08T01:04:42Z","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/2606.07028/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study $L^2$-averaged discrepancies of finite sequences of points in the torus $\\mathbb{T}^d$ with respect to translated and dilated copies of convex bodies with non-smooth boundary. Under suitable anisotropic assumptions on the decay of the Fourier transform of the body, we prove matching lower and upper bounds for the averaged discrepancy, obtaining the rate $ N^{1 - \\frac{d+1}{d^2+d-1}}$. This yields an intermediate regime between smooth convex bodies and polytopes and recovers the known exponent $2/5$ in dimension $d=2$. The argument relies on harmonic analysis techniques combined with a","authors_text":"Alessandro Monguzzi, Luca Brandolini, Roberto Bramati","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-05T08:18:37Z","title":"Discrepancy estimates for multi-dimensional non-smooth convex bodies: a case study"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07028","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":"sha256:7aaf49450996902cd47efd7bfed937b4b28511def95c938bf405118224ec0850","target":"record","created_at":"2026-06-08T01:04:42Z","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":"a15db85fa20b3f53acf7ae603ec635c2e580c80012bbc99c46bcd7a2964f8685","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-06-05T08:18:37Z","title_canon_sha256":"800fafcc82d8ca43a949abd1ceeed4c19fe131f43055544b72c4d78a0c59f9a9"},"schema_version":"1.0","source":{"id":"2606.07028","kind":"arxiv","version":1}},"canonical_sha256":"04c01ba0ded9748b3c89eafaa74e1484edd0b8843c28f21fc3d0296afdd45328","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"04c01ba0ded9748b3c89eafaa74e1484edd0b8843c28f21fc3d0296afdd45328","first_computed_at":"2026-06-08T01:04:42.127582Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:04:42.127582Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+BlcxfEdaLGU+8TdwthUqV6SdjonHHj18mJ/3PKWqD3GsiymzHnNLlH8MST4LjB/4yABP+B/7FCBLbUcXvCFBQ==","signature_status":"signed_v1","signed_at":"2026-06-08T01:04:42.128524Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.07028","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7aaf49450996902cd47efd7bfed937b4b28511def95c938bf405118224ec0850","sha256:0945ef65caa0beeb6994e4053a0d4822952b3bb707fc954fd3f2ce43369e266b"],"state_sha256":"6b4434f73396544c640b143c9b9058ce22f45851284f8b41a68514d640d5c539"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hd6MVxiXpLp8K5NF0xanInntlv7hPlFYmLsJfQuA4RMttrZJmnR8vkIwYo8EtIL8vWQKkcnDL7hro3F4mbOTBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T05:34:32.397058Z","bundle_sha256":"5ade50448fa78a218dc6c2ecf97ed3a7202c58ff92616b5f8acdf0fba023a89f"}}