{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:WS4XOMRNEI4JU3TTBLWG3353EE","short_pith_number":"pith:WS4XOMRN","canonical_record":{"source":{"id":"1612.05152","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-15T17:32:04Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9ef917e64079a14063b237208b7921da9106a0fd303e3a1c58cf014a4692d762","abstract_canon_sha256":"b91f3fcd12cf774788cf66567e20fef937b78084461e6bba1fd03a95e7b0616a"},"schema_version":"1.0"},"canonical_sha256":"b4b977322d22389a6e730aec6defbb2115d6a3c74338c78adee7ad2ae9f224a1","source":{"kind":"arxiv","id":"1612.05152","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.05152","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"arxiv_version","alias_value":"1612.05152v5","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05152","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"pith_short_12","alias_value":"WS4XOMRNEI4J","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WS4XOMRNEI4JU3TT","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WS4XOMRN","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:WS4XOMRNEI4JU3TTBLWG3353EE","target":"record","payload":{"canonical_record":{"source":{"id":"1612.05152","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-15T17:32:04Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9ef917e64079a14063b237208b7921da9106a0fd303e3a1c58cf014a4692d762","abstract_canon_sha256":"b91f3fcd12cf774788cf66567e20fef937b78084461e6bba1fd03a95e7b0616a"},"schema_version":"1.0"},"canonical_sha256":"b4b977322d22389a6e730aec6defbb2115d6a3c74338c78adee7ad2ae9f224a1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:30.681786Z","signature_b64":"2jrTl98TxzcrzLNWlovUELFfDG51lKEeuc26jNB7ZRM0Vwnwu+r4Az81Us5/epeqghKo3gY8ZGOQ1xGv//+ZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4b977322d22389a6e730aec6defbb2115d6a3c74338c78adee7ad2ae9f224a1","last_reissued_at":"2026-05-18T00:01:30.681342Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:30.681342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.05152","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WJ/n/wjG5Dx5M704psMRNyhGAD8SQRX4yT2KMqeN3YFUvnY2oWHnqn0nZ5RV0Xd8gF7IcJ5SbgU+FAKndPCCDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T03:05:54.713904Z"},"content_sha256":"6c3db3019483d8c3f7b0017908d31c11e9920541cdae980e654e78632e1cd62e","schema_version":"1.0","event_id":"sha256:6c3db3019483d8c3f7b0017908d31c11e9920541cdae980e654e78632e1cd62e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:WS4XOMRNEI4JU3TTBLWG3353EE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Properness of nilprogressions and the persistence of polynomial growth of given degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Matthew Tointon, Romain Tessera","submitted_at":"2016-12-15T17:32:04Z","abstract_excerpt":"We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of the Freiman-Bilu result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a Lie-algebra version of the geometry-of-numbers argument at the centre of that result. We also present some applications. We verify a conjecture of Benjamini that if $S$ is a symmetric generating set for a group such that $1\\in S$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05152","kind":"arxiv","version":5},"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":"2026-05-18T00:01:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OhYxniu7M+JW03MT6DjCe1X2u10KQaK3tIQ0u20yLiqX6afYY9y2YEsFM9mEAIEMV3i2h1sg7YOl1GRH4MxMDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T03:05:54.714247Z"},"content_sha256":"1132b36ac5ea9738f78478657af34c26ead34503a752ff9d51655343fabed715","schema_version":"1.0","event_id":"sha256:1132b36ac5ea9738f78478657af34c26ead34503a752ff9d51655343fabed715"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WS4XOMRNEI4JU3TTBLWG3353EE/bundle.json","state_url":"https://pith.science/pith/WS4XOMRNEI4JU3TTBLWG3353EE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WS4XOMRNEI4JU3TTBLWG3353EE/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":"2026-06-24T03:05:54Z","links":{"resolver":"https://pith.science/pith/WS4XOMRNEI4JU3TTBLWG3353EE","bundle":"https://pith.science/pith/WS4XOMRNEI4JU3TTBLWG3353EE/bundle.json","state":"https://pith.science/pith/WS4XOMRNEI4JU3TTBLWG3353EE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WS4XOMRNEI4JU3TTBLWG3353EE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WS4XOMRNEI4JU3TTBLWG3353EE","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":"b91f3fcd12cf774788cf66567e20fef937b78084461e6bba1fd03a95e7b0616a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-15T17:32:04Z","title_canon_sha256":"9ef917e64079a14063b237208b7921da9106a0fd303e3a1c58cf014a4692d762"},"schema_version":"1.0","source":{"id":"1612.05152","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.05152","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"arxiv_version","alias_value":"1612.05152v5","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05152","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"pith_short_12","alias_value":"WS4XOMRNEI4J","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WS4XOMRNEI4JU3TT","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WS4XOMRN","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:1132b36ac5ea9738f78478657af34c26ead34503a752ff9d51655343fabed715","target":"graph","created_at":"2026-05-18T00:01:30Z","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 show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of the Freiman-Bilu result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a Lie-algebra version of the geometry-of-numbers argument at the centre of that result. We also present some applications. We verify a conjecture of Benjamini that if $S$ is a symmetric generating set for a group such that $1\\in S$ and $","authors_text":"Matthew Tointon, Romain Tessera","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-15T17:32:04Z","title":"Properness of nilprogressions and the persistence of polynomial growth of given degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05152","kind":"arxiv","version":5},"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":"sha256:6c3db3019483d8c3f7b0017908d31c11e9920541cdae980e654e78632e1cd62e","target":"record","created_at":"2026-05-18T00:01:30Z","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":"b91f3fcd12cf774788cf66567e20fef937b78084461e6bba1fd03a95e7b0616a","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-15T17:32:04Z","title_canon_sha256":"9ef917e64079a14063b237208b7921da9106a0fd303e3a1c58cf014a4692d762"},"schema_version":"1.0","source":{"id":"1612.05152","kind":"arxiv","version":5}},"canonical_sha256":"b4b977322d22389a6e730aec6defbb2115d6a3c74338c78adee7ad2ae9f224a1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b4b977322d22389a6e730aec6defbb2115d6a3c74338c78adee7ad2ae9f224a1","first_computed_at":"2026-05-18T00:01:30.681342Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:30.681342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2jrTl98TxzcrzLNWlovUELFfDG51lKEeuc26jNB7ZRM0Vwnwu+r4Az81Us5/epeqghKo3gY8ZGOQ1xGv//+ZCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:30.681786Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.05152","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c3db3019483d8c3f7b0017908d31c11e9920541cdae980e654e78632e1cd62e","sha256:1132b36ac5ea9738f78478657af34c26ead34503a752ff9d51655343fabed715"],"state_sha256":"38312ce5bdcb05754c914f76287501b6945875ebff1313b591785f02756d7388"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VdohgUTx5CBf6r7rKdCSB/j7ZWnkRAUiyQ3f9XBMYEHzXSLoPM1MdbGBdN0cW8UyS2FPom2mGAOOYW5q5VU1CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T03:05:54.716155Z","bundle_sha256":"467634bd39295bd6b26123866c681d775db79b45a08e6b34158bab05544212f3"}}