{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:HRQQREMKO2USPH24BS5THZAFGM","short_pith_number":"pith:HRQQREMK","canonical_record":{"source":{"id":"1110.2986","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-13T16:02:35Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7b4389834eb219be6710ff0983d9350a1d9534ba727d7346529e68456ac66787","abstract_canon_sha256":"2eb1414d4ceadbd8ad82482008f6d6a80124df61d9462cdc2893d27469e07c87"},"schema_version":"1.0"},"canonical_sha256":"3c6108918a76a9279f5c0cbb33e4053336f46a4bca354f46db97d45de8004734","source":{"kind":"arxiv","id":"1110.2986","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.2986","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"arxiv_version","alias_value":"1110.2986v3","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2986","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"pith_short_12","alias_value":"HRQQREMKO2US","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HRQQREMKO2USPH24","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HRQQREMK","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:HRQQREMKO2USPH24BS5THZAFGM","target":"record","payload":{"canonical_record":{"source":{"id":"1110.2986","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-13T16:02:35Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7b4389834eb219be6710ff0983d9350a1d9534ba727d7346529e68456ac66787","abstract_canon_sha256":"2eb1414d4ceadbd8ad82482008f6d6a80124df61d9462cdc2893d27469e07c87"},"schema_version":"1.0"},"canonical_sha256":"3c6108918a76a9279f5c0cbb33e4053336f46a4bca354f46db97d45de8004734","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:10.797493Z","signature_b64":"4aUFBlQ1budeb7sz4L//grtj1VKz79t+HA+bbxGw84Pugg+rtqKETeR+ZSjUNorLEPPJaook7CZdIrng5kfcBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c6108918a76a9279f5c0cbb33e4053336f46a4bca354f46db97d45de8004734","last_reissued_at":"2026-05-18T03:45:10.796723Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:10.796723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.2986","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:45:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2kG4MdVHYvUNJimud2H8sFGZwjdRvRtGarI27E1gD1H84wVGTtCHdx3GmMAxiysLX+I6sOzFjkhj29rQE1e6BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T03:01:52.783797Z"},"content_sha256":"47276723d054c155352a4a34285df5538a3d991f1cec4fae1a8ae16485559f49","schema_version":"1.0","event_id":"sha256:47276723d054c155352a4a34285df5538a3d991f1cec4fae1a8ae16485559f49"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:HRQQREMKO2USPH24BS5THZAFGM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Higher moments of convolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Ilya D. Shkredov, Tomasz Schoen","submitted_at":"2011-10-13T16:02:35Z","abstract_excerpt":"We study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In our investigation we use different approaches including basic combinatorics, Fourier analysis and eigenvalues method to establish basic properties of higher energies. We provide also a sequence of applications of higher energies additive combinatorics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2986","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":""},"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":"2026-05-18T03:45:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QnGsWoF8HwN0ds4rZ8sNa/tC7zjbK7HPBUQ9OxzFq9NX+Jducg58fvB9/AmmUfz2KpbdhQeYwQT5LyL7TN0FAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T03:01:52.784466Z"},"content_sha256":"7cdffe0ac1e594bf80edda1b8822a423807a89563683bb5dcd8052f9c3ee638d","schema_version":"1.0","event_id":"sha256:7cdffe0ac1e594bf80edda1b8822a423807a89563683bb5dcd8052f9c3ee638d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HRQQREMKO2USPH24BS5THZAFGM/bundle.json","state_url":"https://pith.science/pith/HRQQREMKO2USPH24BS5THZAFGM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HRQQREMKO2USPH24BS5THZAFGM/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":"2026-06-10T03:01:52Z","links":{"resolver":"https://pith.science/pith/HRQQREMKO2USPH24BS5THZAFGM","bundle":"https://pith.science/pith/HRQQREMKO2USPH24BS5THZAFGM/bundle.json","state":"https://pith.science/pith/HRQQREMKO2USPH24BS5THZAFGM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HRQQREMKO2USPH24BS5THZAFGM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:HRQQREMKO2USPH24BS5THZAFGM","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":"2eb1414d4ceadbd8ad82482008f6d6a80124df61d9462cdc2893d27469e07c87","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-13T16:02:35Z","title_canon_sha256":"7b4389834eb219be6710ff0983d9350a1d9534ba727d7346529e68456ac66787"},"schema_version":"1.0","source":{"id":"1110.2986","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.2986","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"arxiv_version","alias_value":"1110.2986v3","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2986","created_at":"2026-05-18T03:45:10Z"},{"alias_kind":"pith_short_12","alias_value":"HRQQREMKO2US","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HRQQREMKO2USPH24","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HRQQREMK","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:7cdffe0ac1e594bf80edda1b8822a423807a89563683bb5dcd8052f9c3ee638d","target":"graph","created_at":"2026-05-18T03:45:10Z","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 study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In our investigation we use different approaches including basic combinatorics, Fourier analysis and eigenvalues method to establish basic properties of higher energies. We provide also a sequence of applications of higher energies additive combinatorics.","authors_text":"Ilya D. Shkredov, Tomasz Schoen","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-13T16:02:35Z","title":"Higher moments of convolutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2986","kind":"arxiv","version":3},"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":"sha256:47276723d054c155352a4a34285df5538a3d991f1cec4fae1a8ae16485559f49","target":"record","created_at":"2026-05-18T03:45:10Z","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":"2eb1414d4ceadbd8ad82482008f6d6a80124df61d9462cdc2893d27469e07c87","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-13T16:02:35Z","title_canon_sha256":"7b4389834eb219be6710ff0983d9350a1d9534ba727d7346529e68456ac66787"},"schema_version":"1.0","source":{"id":"1110.2986","kind":"arxiv","version":3}},"canonical_sha256":"3c6108918a76a9279f5c0cbb33e4053336f46a4bca354f46db97d45de8004734","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c6108918a76a9279f5c0cbb33e4053336f46a4bca354f46db97d45de8004734","first_computed_at":"2026-05-18T03:45:10.796723Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:10.796723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4aUFBlQ1budeb7sz4L//grtj1VKz79t+HA+bbxGw84Pugg+rtqKETeR+ZSjUNorLEPPJaook7CZdIrng5kfcBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:10.797493Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.2986","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47276723d054c155352a4a34285df5538a3d991f1cec4fae1a8ae16485559f49","sha256:7cdffe0ac1e594bf80edda1b8822a423807a89563683bb5dcd8052f9c3ee638d"],"state_sha256":"0baa7c5520a9334732a94bf84cb5a7c893c50ecab7950126926ca0c3b880ff9b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XK4BOTWgLAiNPKxtOK3aJkcSwGUjpM3hk86AqJ03wm/S4g4BaMF6ICv1LUpDxHvQQGCbusBewTxRqVGlpLE8Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T03:01:52.788397Z","bundle_sha256":"dc6be61bb57ea1e1803d626fc63c4634f36c143a3a5da74b96acd65bffa24b06"}}