Geodesic
On a Diophantine inequality with reciprocals
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 144-154

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A sharpened lower bound is obtained for the number of solutions to an inequality of the form $\alpha \le \{(a\overline {n}+bn)/q\}\beta $, $1\le n\le N$, where $q$ is a sufficiently large prime number, $a$ and $b$ are integers with $(ab,q)=1$, $n\overline {n}\equiv 1 \pmod q$, and $0\le \alpha \beta \le 1$. The length $N$ of the range of the variable $n$ is of order $q^\varepsilon $, where $\varepsilon >0$ is an arbitrarily small fixed number.
Mots-clés : inverse residues
Keywords: fractional parts, Kloosterman sums. 
@article{TM_2017_299_a8,
     author = {M. A. Korolev},
     title = {On a {Diophantine} inequality with reciprocals},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {144--154},
     publisher = {mathdoc},
     volume = {299},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_299_a8/}
}
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M. A. Korolev. On a Diophantine inequality with reciprocals. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 144-154. http://geodesic.mathdoc.fr/item/TM_2017_299_a8/