{"paper":{"title":"Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime 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":"2014-04-20T20:39:09Z","abstract_excerpt":"In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\\equiv y r\\pmod p;\\quad x,y\\in \\mathbb{N},\\quad x,y\\le H,\\quad r\\in\\cU, $$ for certain ranges of $H$ and $|\\cU|$, where $\\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$ x^n\\equiv \\lambda\\pmod p; \\quad x\\in \\N, \\quad L<x<L+p/n, $$ is at most $p^{\\frac{1}{3}-c}$ uniformly over positive integers $n, \\lambda$ and $L$, for some absolute constant $c>0$. This impl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5070","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"}