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This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $(\\log p)^2$ steps suffice for $X_n$ to be close to uniformly distributed on the integers mod $p$ for all odd $p$ while order $(\\log p)^2$ steps are necessary for $X_n$ to be close to uniformly"},"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":"2007.09126","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-07-17T17:35:13Z","cross_cats_sorted":[],"title_canon_sha256":"4453c4750c975b79bcd6fe11e74fa3dcbb318c96dd7c71bef7c7f12dd57911ac","abstract_canon_sha256":"0c1fc06ffbd46850f3509aa81a574db0c343c0d581d0b8d9cc74a1faae213cf6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:27:23.417563Z","signature_b64":"FPyVMIbqz8rBGqgVpZg8ygNqKcRzhT3nR2yKCVBExC8VJ8H55kTOAnELD36zPlrEZEQVA8LdNckWIMQA7xGVAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71cf1ca28872cc8421bfd22ae7d72cea738bdce044a715f6355b43b63be08b82","last_reissued_at":"2026-07-05T02:27:23.417132Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:27:23.417132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A multiplicatively symmetrized version of the Chung-Diaconis-Graham random process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Martin Hildebrand (University at Albany, State University of New York)","submitted_at":"2020-07-17T17:35:13Z","abstract_excerpt":"This paper considers random processes of the form $X_{n+1}=a_nX_n+b_n \\pmod p$ where $p$ is odd, $X_0=0$, $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d., and $a_n$ and $b_n$ are independent with $P(a_n=2)=P(a_n=(p+1)/2)=1/2$ and $P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$. This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $(\\log p)^2$ steps suffice for $X_n$ to be close to uniformly distributed on the integers mod $p$ for all odd $p$ while order $(\\log p)^2$ steps are necessary for $X_n$ to be close to uniformly"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.09126","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/2007.09126/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2007.09126","created_at":"2026-07-05T02:27:23.417186+00:00"},{"alias_kind":"arxiv_version","alias_value":"2007.09126v3","created_at":"2026-07-05T02:27:23.417186+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.09126","created_at":"2026-07-05T02:27:23.417186+00:00"},{"alias_kind":"pith_short_12","alias_value":"OHHRZIUIOLGI","created_at":"2026-07-05T02:27:23.417186+00:00"},{"alias_kind":"pith_short_16","alias_value":"OHHRZIUIOLGIIIN7","created_at":"2026-07-05T02:27:23.417186+00:00"},{"alias_kind":"pith_short_8","alias_value":"OHHRZIUI","created_at":"2026-07-05T02:27:23.417186+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/OHHRZIUIOLGIIIN72IVOPVZM5J","json":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J.json","graph_json":"https://pith.science/api/pith-number/OHHRZIUIOLGIIIN72IVOPVZM5J/graph.json","events_json":"https://pith.science/api/pith-number/OHHRZIUIOLGIIIN72IVOPVZM5J/events.json","paper":"https://pith.science/paper/OHHRZIUI"},"agent_actions":{"view_html":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J","download_json":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J.json","view_paper":"https://pith.science/paper/OHHRZIUI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2007.09126&json=true","fetch_graph":"https://pith.science/api/pith-number/OHHRZIUIOLGIIIN72IVOPVZM5J/graph.json","fetch_events":"https://pith.science/api/pith-number/OHHRZIUIOLGIIIN72IVOPVZM5J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J/action/storage_attestation","attest_author":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J/action/author_attestation","sign_citation":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J/action/citation_signature","submit_replication":"https://pith.science/pith/OHHRZIUIOLGIIIN72IVOPVZM5J/action/replication_record"}},"created_at":"2026-07-05T02:27:23.417186+00:00","updated_at":"2026-07-05T02:27:23.417186+00:00"}