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The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent.\n  Answering an old question of Mycielski we show that if $G$ is not locally compact then there exists a Borel Haar null set that is not contained in any $G_\\delta$ Haar null set. We also show that $G_\\delta$ can be replaced by an"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.7667","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2013-12-30T09:42:33Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"08edcc4b9a0875fc48b397977222e344e41a56f3a7b45a1aabc30d138cb89517","abstract_canon_sha256":"6de2c804f2178088406b3b83665f6e94fefb9760dfdf8f8ba1b1a694b0b0eb55"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:20.953545Z","signature_b64":"py9yCVek0POX6RfbSjNKyTp3wUrp2vQe+Ec/pNM63+xJ2Gos6ek3+L6NRWp0qPAaKRcim9VV5TqkHFkdeSfZBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93d6fc8955a755a5a39bb08e58a1075cf792d97672982df9aac883b429502896","last_reissued_at":"2026-05-18T01:23:20.952898Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:20.952898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Haar null sets without $G_\\delta$ hulls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.LO","authors_text":"M\\'arton Elekes, Zolt\\'an Vidny\\'anszky","submitted_at":"2013-12-30T09:42:33Z","abstract_excerpt":"Let $G$ be an abelian Polish group, e.g. a separable Banach space. A subset $X \\subset G$ is called Haar null (in the sense of Christensen) if there exists a Borel set $B \\supset X$ and a Borel probability measure $\\mu$ on $G$ such that $\\mu(B+g)=0$ for every $g \\in G$. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent.\n  Answering an old question of Mycielski we show that if $G$ is not locally compact then there exists a Borel Haar null set that is not contained in any $G_\\delta$ Haar null set. We also show that $G_\\delta$ can be replaced by an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7667","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1312.7667","created_at":"2026-05-18T01:23:20.952995+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.7667v4","created_at":"2026-05-18T01:23:20.952995+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.7667","created_at":"2026-05-18T01:23:20.952995+00:00"},{"alias_kind":"pith_short_12","alias_value":"SPLPZCKVU5K2","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_16","alias_value":"SPLPZCKVU5K2LI43","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_8","alias_value":"SPLPZCKV","created_at":"2026-05-18T12:27:59.945178+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT","json":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT.json","graph_json":"https://pith.science/api/pith-number/SPLPZCKVU5K2LI43WCHFRIIHLT/graph.json","events_json":"https://pith.science/api/pith-number/SPLPZCKVU5K2LI43WCHFRIIHLT/events.json","paper":"https://pith.science/paper/SPLPZCKV"},"agent_actions":{"view_html":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT","download_json":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT.json","view_paper":"https://pith.science/paper/SPLPZCKV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.7667&json=true","fetch_graph":"https://pith.science/api/pith-number/SPLPZCKVU5K2LI43WCHFRIIHLT/graph.json","fetch_events":"https://pith.science/api/pith-number/SPLPZCKVU5K2LI43WCHFRIIHLT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT/action/storage_attestation","attest_author":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT/action/author_attestation","sign_citation":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT/action/citation_signature","submit_replication":"https://pith.science/pith/SPLPZCKVU5K2LI43WCHFRIIHLT/action/replication_record"}},"created_at":"2026-05-18T01:23:20.952995+00:00","updated_at":"2026-05-18T01:23:20.952995+00:00"}