{"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"}