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Ann."],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{GL}(n)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>GL<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for arbitrary <jats:inline-formula><jats:alternatives><jats:tex-math>$$n \\geqslant 3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>\u2a7e<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as exponential sums. As an application we go beyond Sarnak\u2019s density conjecture for the principal congruence subgroup of prime level. We also obtain power-saving bounds for all Kloosterman sums on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{GL}(4)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>GL<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00208-023-02777-6","type":"journal-article","created":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T22:02:36Z","timestamp":1703714556000},"page":"1171-1200","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bounds for Kloosterman sums on $$\\textrm{GL}(n)$$"],"prefix":"10.1007","volume":"390","author":[{"given":"Valentin","family":"Blomer","sequence":"first","affiliation":[]},{"given":"Siu Hang","family":"Man","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,12,27]]},"reference":[{"key":"2777_CR1","unstructured":"Assing, E., Blomer, V.: The density conjecture for principal congruence subgroups. 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