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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 "},"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":"1810.06341","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-15T13:23:40Z","cross_cats_sorted":[],"title_canon_sha256":"158a3d0c6c8357902bd82d980460fe25f8ee6ab6830fada44727acb80472606e","abstract_canon_sha256":"7701f5e2ea3e61c1368c4d1722626c155d4dff569a86581af9d7935bf5332af7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:20.769074Z","signature_b64":"0HN1gI3pm8vNdQGnWga1pL8LcCcqVxa3RQojN/0RvNHNIf0kH54oozrlDM+N+n0pS9Yn+/a+PhxBhtZF3UwbCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7546ad702f82d560b2980a114ee5b9de30496e981cb4f44c603afe85c14605ae","last_reissued_at":"2026-05-18T00:03:20.768630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:20.768630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $|S|< N^{2-1/8 + o(1)}$. 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