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Sets of the form $\mathscr A+\mathscr B$ and finite continued fractions
Sbornik. Mathematics, Tome 198 (2007) no. 4, pp. 537-557 Cet article a éte moissonné depuis la source Math-Net.Ru

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Estimates are obtained for the number of proper irreducible fractions with denominator $p$ such that an initial and an end segment of their expansion in a continued fraction have bounded partial quotients. These results are connected with an estimate of incomplete Kloosterman sums over sets of the form $\mathscr A+\mathscr B\subset\mathbb Z_p$. Results on the distribution in $\mathbb Z_p$ of the elements of sets of the form $(\mathscr A+\mathscr B)^k$ and $k\cdot(\mathscr A+\mathscr B)^{-1}$ are obtained. Bibliography: 21 titles. 
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N. G. Moshchevitin. Sets of the form $\mathscr A+\mathscr B$ and finite continued fractions. Sbornik. Mathematics, Tome 198 (2007) no. 4, pp. 537-557. http://geodesic.mathdoc.fr/item/SM_2007_198_4_a4/

[1] N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, Fizmatgiz, M., 1963 | MR | Zbl

[2] N. M. Korobov, “O nekotorykh voprosakh teorii diofantovykh priblizhenii”, UMN, 22:3 (1967), 83–118 | MR | Zbl

[3] S. K. Zaremba, “La méthode des “bons treillis” pour le calcul des intégrales multiples”, Applications of number theory to numerical analysis, Proc. Sympos. Univ. Montreal (Montreal, QC, 1971), Academic Press, New York, 1972, 39–119 | MR | Zbl

[4] T. W. Cusick, “Zaremba's conjecture and sums of the divisor function”, Math. Comp., 61:203 (1993), 171–176 | DOI | MR | Zbl

[5] T. W. Cusick, “Products of simultaneous approximations of rational numbers”, Arch. Math. (Basel), 53:2 (1989), 154–158 | DOI | MR | Zbl

[6] H. Niederreiter, “Dyadic fractions with small partial quotients”, Monatsh. Math., 101:4 (1986), 309–315 | DOI | MR | Zbl

[7] A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent, Hermann Cie, Paris, 1948 | MR | Zbl

[8] A. Weil, “On some exponential sums”, Proc. Natl. Acad. Sci. USA, 34:5 (1948), 204–207 | DOI | MR | Zbl

[9] D. Hensley, “Continued fraction Cantor sets, Hausdorff dimension, and functional analysis”, J. Number Theory, 40:3 (1992), 336–358 | DOI | MR | Zbl

[10] D. Hensley, “The distribution of badly approximable numbers and continuants with bounded digits”, Théorie des nombres (Québec, PQ, 1987), de Gruyter, Berlin, 1989, 371–385 | MR | Zbl

[11] D. Hensley, “The distribution of badly approximable rationals and continuants with bounded digits, II”, J. Number Theory, 34:3 (1990), 293–334 | DOI | MR | Zbl

[12] S. V. Konyagin, N. G. Moschevitin, “Predstavlenie ratsionalnykh chisel konechnymi tsepnymi drobyami”, UMN, 51:4 (1996), 159–160 | MR | Zbl

[13] N. G. Moschevitin, “O tsepnykh drobyakh i pochti arifmeticheskikh progressiyakh”, UMN, 55:1 (2000), 189–190 | MR | Zbl

[14] L. Keipers, G. Niderraiter, Ravnomernoe raspredelenie posledovatelnostei, Nauka, M., 1985 ; L. Kuipers, H. Niederreiter, Uniform distribution of sequences, Pure Appl. Math., Wiley-Intersci., New York–London–Sydney, 1974 | MR | Zbl | MR | Zbl

[15] R. Lidl, G. Niderraiter, Konechnye polya, t. 1, Mir, M., 1988 ; R. Lidl, H. Niederreiter, Finite fields, Encyclopedia Math. Appl., 20, Addison-Wesley, Reading, MA, 1983 | MR | Zbl | MR | Zbl

[16] S. A. Stepanov, Arifmetika algebraicheskikh krivykh, Nauka, M., 1991 ; S. A. Stepanov, Arithmetic of algebraic curves, Monogr. Contemp. Math., Consultants Bureau, New York, 1994 | MR | Zbl | MR | Zbl

[17] S. A. Stepanov, “Ob otsenke ratsionalnykh trigonometricheskikh summ s prostym znamenatelem”, Sbornik statei. Posvyaschaetsya akademiku I. M. Vinogradovu k ego 80-letiyu. I, Tr. MIAN, 112, Nauka, M., 1971, 346–371 ; S. A. Stepanov, “On estimating rational trigonometric sums with prime denominator”, Proc. Steklov Inst. Math., 112 (1973), 358–385 | MR | Zbl | Zbl

[18] N. M. Korobov, Trigonometricheskie summy i ikh prilozheniya, Nauka, M., 1989 ; N. M. Korobov, Exponential sums and their applications, Math. Appl. (Soviet Ser.), 80, Kluwer Acad. Publ., Dordrecht, 1992 | MR | Zbl | MR | Zbl

[19] S. V. Konyagin, “Otsenki trigonometricheskikh summ po podgruppam i summ Gaussa”, IV Mezhdunarodnaya konferentsiya “Sovremennye problemy teorii chisel i ee prilozheniya”, posvyaschennaya 180-letiyu P. L. Chebysheva i 110-letiyu I. M. Vinogradova: Aktualnye problemy, ch. III (Tula, 2001), MGU, mekhmat, 2002, 86–114 | MR

[20] W. D. Banks, A. Conflitti, I. E. Shparlinski, “Character sums over integers with restricted $g$-ary digits”, Illinois J. Math, 46:3 (2002), 819–836 | MR | Zbl

[21] A. A. Karatsuba, “Summy kharakterov s prostymi chislami”, Izv. AN SSSR. Ser. matem., 34:2 (1970), 299–321 | MR | Zbl