{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:GLXQC55C6ENVKEPLGOVJZI3ZWJ","short_pith_number":"pith:GLXQC55C","canonical_record":{"source":{"id":"1802.09066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-25T19:23:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"897ebf421ee2ba32f506ad932fad95fdc3175107038ede33ee0767b25d9e1cb9","abstract_canon_sha256":"ce3b62492f6a196815c1e63e3f315217068bdd4fa8523d01c27a19b77c6ea704"},"schema_version":"1.0"},"canonical_sha256":"32ef0177a2f11b5511eb33aa9ca379b244abe514291fe647663132bba12a6cd3","source":{"kind":"arxiv","id":"1802.09066","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.09066","created_at":"2026-05-18T00:22:02Z"},{"alias_kind":"arxiv_version","alias_value":"1802.09066v2","created_at":"2026-05-18T00:22:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.09066","created_at":"2026-05-18T00:22:02Z"},{"alias_kind":"pith_short_12","alias_value":"GLXQC55C6ENV","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GLXQC55C6ENVKEPL","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GLXQC55C","created_at":"2026-05-18T12:32:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:GLXQC55C6ENVKEPLGOVJZI3ZWJ","target":"record","payload":{"canonical_record":{"source":{"id":"1802.09066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-25T19:23:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"897ebf421ee2ba32f506ad932fad95fdc3175107038ede33ee0767b25d9e1cb9","abstract_canon_sha256":"ce3b62492f6a196815c1e63e3f315217068bdd4fa8523d01c27a19b77c6ea704"},"schema_version":"1.0"},"canonical_sha256":"32ef0177a2f11b5511eb33aa9ca379b244abe514291fe647663132bba12a6cd3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:02.119912Z","signature_b64":"b2L3iP8GpLHZRX+zhLnPkHhCB8JaCYnBw9nXK2k9/KrSygEmexASuuKmHpazihLwSP+d2H/FOhCdqJT4sZUzBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32ef0177a2f11b5511eb33aa9ca379b244abe514291fe647663132bba12a6cd3","last_reissued_at":"2026-05-18T00:22:02.119462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:02.119462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.09066","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0msXXD6NZPp+AY7TdIKk3HcctqpWmh5NTARlnNTxeBoIdBo4O3Ay7Ts/fM6X9fXTShyD7jZMHgne+BndaVXQDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T02:47:18.748677Z"},"content_sha256":"c780489dbb88b08467cf25d48b680a653f516a7a1b79ddbd6cabe2afbf11ba0b","schema_version":"1.0","event_id":"sha256:c780489dbb88b08467cf25d48b680a653f516a7a1b79ddbd6cabe2afbf11ba0b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:GLXQC55C6ENVKEPLGOVJZI3ZWJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On asymptotic formulae in some sum-product questions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ilya D. Shkredov","submitted_at":"2018-02-25T19:23:08Z","abstract_excerpt":"In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\\mathbf{F}_p$, as well as the growth result in ${\\rm SL}_2 (\\mathbf{F}_p)$ due to Helfgott. Here some of our applications:\n  $\\bullet~$ a new bound for the number of the solutions to the equation $(a_1-a_2) (a_3-a_4) = (a'_1-a'_2) (a'_3-a'_4)$, $\\,a_i, a'_i\\in A$, $A$ is an arbitrary subset of $\\mathbf{F}_p$,\n  $\\bullet~$ a new effective bound for multilinear exponential sums of Bourgain,\n  $\\bullet~$ an asymptotic analogue"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09066","kind":"arxiv","version":2},"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":"2026-05-18T00:22:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tEMCk2QkihEbMIFCX9e1mhBaTPphmO1z6sMEe9YxSqa+cCwOFK6RG6ut7oxQjvSsD7HYjLj+36QapttLqttXAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T02:47:18.749375Z"},"content_sha256":"3b82a6aaf795316c4b24f10bdb725a0e16fb09a74ecfac9272468fc10c92dec8","schema_version":"1.0","event_id":"sha256:3b82a6aaf795316c4b24f10bdb725a0e16fb09a74ecfac9272468fc10c92dec8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GLXQC55C6ENVKEPLGOVJZI3ZWJ/bundle.json","state_url":"https://pith.science/pith/GLXQC55C6ENVKEPLGOVJZI3ZWJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GLXQC55C6ENVKEPLGOVJZI3ZWJ/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":"2026-06-10T02:47:18Z","links":{"resolver":"https://pith.science/pith/GLXQC55C6ENVKEPLGOVJZI3ZWJ","bundle":"https://pith.science/pith/GLXQC55C6ENVKEPLGOVJZI3ZWJ/bundle.json","state":"https://pith.science/pith/GLXQC55C6ENVKEPLGOVJZI3ZWJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GLXQC55C6ENVKEPLGOVJZI3ZWJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:GLXQC55C6ENVKEPLGOVJZI3ZWJ","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":"ce3b62492f6a196815c1e63e3f315217068bdd4fa8523d01c27a19b77c6ea704","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-25T19:23:08Z","title_canon_sha256":"897ebf421ee2ba32f506ad932fad95fdc3175107038ede33ee0767b25d9e1cb9"},"schema_version":"1.0","source":{"id":"1802.09066","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.09066","created_at":"2026-05-18T00:22:02Z"},{"alias_kind":"arxiv_version","alias_value":"1802.09066v2","created_at":"2026-05-18T00:22:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.09066","created_at":"2026-05-18T00:22:02Z"},{"alias_kind":"pith_short_12","alias_value":"GLXQC55C6ENV","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GLXQC55C6ENVKEPL","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GLXQC55C","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:3b82a6aaf795316c4b24f10bdb725a0e16fb09a74ecfac9272468fc10c92dec8","target":"graph","created_at":"2026-05-18T00:22:02Z","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":"In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\\mathbf{F}_p$, as well as the growth result in ${\\rm SL}_2 (\\mathbf{F}_p)$ due to Helfgott. Here some of our applications:\n  $\\bullet~$ a new bound for the number of the solutions to the equation $(a_1-a_2) (a_3-a_4) = (a'_1-a'_2) (a'_3-a'_4)$, $\\,a_i, a'_i\\in A$, $A$ is an arbitrary subset of $\\mathbf{F}_p$,\n  $\\bullet~$ a new effective bound for multilinear exponential sums of Bourgain,\n  $\\bullet~$ an asymptotic analogue","authors_text":"Ilya D. Shkredov","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-25T19:23:08Z","title":"On asymptotic formulae in some sum-product questions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09066","kind":"arxiv","version":2},"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:c780489dbb88b08467cf25d48b680a653f516a7a1b79ddbd6cabe2afbf11ba0b","target":"record","created_at":"2026-05-18T00:22:02Z","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":"ce3b62492f6a196815c1e63e3f315217068bdd4fa8523d01c27a19b77c6ea704","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-25T19:23:08Z","title_canon_sha256":"897ebf421ee2ba32f506ad932fad95fdc3175107038ede33ee0767b25d9e1cb9"},"schema_version":"1.0","source":{"id":"1802.09066","kind":"arxiv","version":2}},"canonical_sha256":"32ef0177a2f11b5511eb33aa9ca379b244abe514291fe647663132bba12a6cd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32ef0177a2f11b5511eb33aa9ca379b244abe514291fe647663132bba12a6cd3","first_computed_at":"2026-05-18T00:22:02.119462Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:02.119462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b2L3iP8GpLHZRX+zhLnPkHhCB8JaCYnBw9nXK2k9/KrSygEmexASuuKmHpazihLwSP+d2H/FOhCdqJT4sZUzBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:02.119912Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.09066","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c780489dbb88b08467cf25d48b680a653f516a7a1b79ddbd6cabe2afbf11ba0b","sha256:3b82a6aaf795316c4b24f10bdb725a0e16fb09a74ecfac9272468fc10c92dec8"],"state_sha256":"51288ad1225c144ea733e8b7b17920807db96251f7cd73faa43439348ba256eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q8nYa+xnvsDBKQB51AWo/TizFYbwLFqqLSlFIiSWoIVkTc97Q6Uo3TRKUe54Is0ndIHnmx+++IjhxwF24kBOBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T02:47:18.753924Z","bundle_sha256":"b30f6bed4173e34ec2d15422127a28049e9696ac2c5557654b02852b28405a86"}}