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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"},"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":"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"},"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"},"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. 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