{"paper":{"title":"Double exponential sums and congruences with intervals and exponential functions modulo a prime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"M. Z. Garaev","submitted_at":"2018-10-15T13:23:40Z","abstract_excerpt":"Let $p$ be a large prime number and $g$ be any integer of multiplicative order $T$ modulo $p$. We obtain a new estimate of the double exponential sum $$ S=\\sum_{n\\in \\mathcal{N}}\\left|\\sum_{m\\in \\mathcal{M} }e_p(an g^{m})\\right|, \\quad \\gcd (a,p)=1, $$ where $\\mathcal{N}$ and $\\mathcal{M}$ are intervals of consecutive integers with $|\\mathcal{N}|=N$ and $|\\mathcal{M}|=M<T$ elements. One representative example is the following consequence of the main result: if $N=M\\approx p^{1/3}$, then $|S|< N^{2-1/8 + o(1)}$. We then apply our estimate to obtain new results on additive congruences involving "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06341","kind":"arxiv","version":1},"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"}