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Garaev","submitted_at":"2010-07-09T08:04:03Z","abstract_excerpt":"Let $p$ be a large prime number, $K,L,M,\\lambda$ be integers with $1\\le M\\le p$ and ${\\color{red}\\gcd}(\\lambda,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$ xy\\equiv\\lambda \\pmod p, \\qquad K+1\\le x\\le K+M,\\quad L+1\\le y\\le L+M $$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\\equiv\\lambda\\pmod p, \\quad L+1\\le x,y,z\\le L+M. $$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\\le M^{"},"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":"1007.1526","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-09T08:04:03Z","cross_cats_sorted":[],"title_canon_sha256":"fa0e26ac9d2d3cb0a4b3ac8e9b2ddd258bf89352fb01241a0b8573a0a7421063","abstract_canon_sha256":"cf46b2b35b5a2cb3b4a7b67c12df33d588031b845edd016c256c08b5df9e39b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:27.640849Z","signature_b64":"LyJIEXP3NuJtZcsNsxl1bQaG1b9bfoaDjcRZYCi1P2URM9wC1dVsVw8R1CyeIuwK8jPi61BfbmRccavBxnqXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3685be7e5f4b9c0922a6495f326a054a1e828c128c819feb0f51d8da3c66fb16","last_reissued_at":"2026-05-18T04:39:27.640409Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:27.640409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Concentration points on two and three dimensional modular hyperbolas and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"J. Cilleruelo, M. Z. Garaev","submitted_at":"2010-07-09T08:04:03Z","abstract_excerpt":"Let $p$ be a large prime number, $K,L,M,\\lambda$ be integers with $1\\le M\\le p$ and ${\\color{red}\\gcd}(\\lambda,p)=1.$ The aim of our paper is to obtain sharp upper bound estimates for the number $I_2(M; K,L)$ of solutions of the congruence $$ xy\\equiv\\lambda \\pmod p, \\qquad K+1\\le x\\le K+M,\\quad L+1\\le y\\le L+M $$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\\equiv\\lambda\\pmod p, \\quad L+1\\le x,y,z\\le L+M. $$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\\le M^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1526","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1007.1526","created_at":"2026-05-18T04:39:27.640473+00:00"},{"alias_kind":"arxiv_version","alias_value":"1007.1526v2","created_at":"2026-05-18T04:39:27.640473+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1526","created_at":"2026-05-18T04:39:27.640473+00:00"},{"alias_kind":"pith_short_12","alias_value":"G2C347S7JOOA","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_16","alias_value":"G2C347S7JOOASIVG","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_8","alias_value":"G2C347S7","created_at":"2026-05-18T12:26:07.630475+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/G2C347S7JOOASIVGJFPTE2QFJI","json":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI.json","graph_json":"https://pith.science/api/pith-number/G2C347S7JOOASIVGJFPTE2QFJI/graph.json","events_json":"https://pith.science/api/pith-number/G2C347S7JOOASIVGJFPTE2QFJI/events.json","paper":"https://pith.science/paper/G2C347S7"},"agent_actions":{"view_html":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI","download_json":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI.json","view_paper":"https://pith.science/paper/G2C347S7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1007.1526&json=true","fetch_graph":"https://pith.science/api/pith-number/G2C347S7JOOASIVGJFPTE2QFJI/graph.json","fetch_events":"https://pith.science/api/pith-number/G2C347S7JOOASIVGJFPTE2QFJI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/action/storage_attestation","attest_author":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/action/author_attestation","sign_citation":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/action/citation_signature","submit_replication":"https://pith.science/pith/G2C347S7JOOASIVGJFPTE2QFJI/action/replication_record"}},"created_at":"2026-05-18T04:39:27.640473+00:00","updated_at":"2026-05-18T04:39:27.640473+00:00"}