×

Modular hyperbolas. (English) Zbl 1272.11050

The author gives a survey of recent results on the distribution of points \((x,y)\) on modular hyperbolas \(xy \equiv a\pmod m\). The article contains a very diverse range of applications of such results (geometric properties of modular hyperbolas, Lehmer problem, arithmetic structure of shifted products, sums of divisor functions over quadratic polynomials and arithmetic progressions, properties of elliptic curves, Frobenius numbers, coefficients of cyclotomic polynomials, continued fractions, discrete logarithm algorithms, multivariate generalizations, etc).
The author suggests a number of open problems of different levels of difficulty. The paper also can be considered as a good source of links concerning applications of Kloosterman sums.

MSC:

11D79 Congruences in many variables
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11L05 Gauss and Kloosterman sums; generalizations
11L07 Estimates on exponential sums
11L40 Estimates on character sums
11N69 Distribution of integers in special residue classes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahmadi O., Shparlinski I.E.: Distribution of matrices with restricted entries over finite fields, Indag. Math. (N.S.) 18, 327–337 (2007) · Zbl 1181.11030
[2] Alford W.R., Granville A., Pomerance C.: There are infinitely many Carmichael numbers. Ann. of Math. 39(2), 703–722 (1994) · Zbl 0816.11005
[3] Aliev I.M., Gruber P.M.: An optimal lower bound for the Frobenius problem. J. Number Theory 123, 71–79 (2007) · Zbl 1114.11025
[4] Aliev I.M., Henk M., Hinrichs A.: Frobenius numbers. J. Combin. Theory Ser. A 118, 525–531 (2011) · Zbl 1237.11013
[5] Alkan E., Stan F., Zaharescu A.: Lehmer k-tuples. Proc. Amer. Math. Soc. 134, 2807–2815 (2006) · Zbl 1173.11045
[6] Alkan E., Xiong M., Zaharescu A.: Quotients of values of the Dedekind eta function. Math. Ann. 342, 157–176 (2008) · Zbl 1144.11034
[7] Andrews G.E.: A lower bound for the volume of strictly convex bodies with many boundary lattice points. Trans. Amer. Math. Soc. 106, 270–279 (1963) · Zbl 0118.28301
[8] V.I. Arnold, Statistics of integral convex polygons, Funktsional. Anal. i Prilozhen., 14 (1980), no. 2, 1–3 (in Russian).
[9] M.O. Avdeeva, On the statistics of partial quotients of finite continued fractions, Funktsional. Anal. i Prilozhen, 38 (2004), no. 2, 1–11 (in Russian); transl. as Funct. Anal. Appl.
[10] Ayyad A.: The distribution of solutions of the congruence $${x_1x_2x_3\(\backslash\)ldots x_n \(\backslash\)equiv c\(\backslash\);(\(\backslash\)mod p)}$$ . Proc. Amer. Math. Soc. 127, 943–950 (1999) · Zbl 0919.11029
[11] Ayyad A., Cochrane T.: Lattices in $${\(\backslash\)mathbb{Z}\^2}$$ and the congruence $${xy+uv\(\backslash\)equiv c\(\backslash\);(\(\backslash\)mod m)}$$ . Acta Arith. 132, 127–133 (2008) · Zbl 1153.11002
[12] Ayyad A., Cochrane T., Zheng Z.: The congruence $${x_1x_2 \(\backslash\)equiv x_3x_4\(\backslash\);(\(\backslash\)mod p)}$$ , the equation x 1 x 2 = x 3 x 4 and the mean value of character sums. J. Number Theory 59, 398–413 (1996) · Zbl 0869.11003
[13] S. Baier, Multiplicative inverses in short intervals, preprint, 2012, arXiv: 1208.3393.
[14] R.C. Baker, Kloosterman sums with prime variable, preprint, 2011.
[15] A. Balog and J.-M. Deshouillers, On some convex lattice polytopes, In: Number Theory in Progress, 2, de Gruyter, Berlin, 1999, pp. 591–606. · Zbl 0931.52005
[16] Banks W.D., Heath-Brown D.R., Shparlinski I.E.: On the average value of divisor sums in arithmetic progressions. Int. Math. Res. Not. 2005, 1–25 (2005) · Zbl 1071.11055
[17] Bárány I., Pach J.: On the number of convex lattice polygons. Combin. Probab. Comput. 1, 295–302 (1992) · Zbl 0798.52012
[18] Beck J., Khan M.R.: On the uniform distribution of inverses modulo n. Period. Math. Hungar. 44, 147–155 (2002) · Zbl 1017.11043
[19] Beck M., Einstein D., Zacks S.: Some experimental results on the Frobenius problem, Experiment. Math. 12, 263–269 (2003) · Zbl 1076.11015
[20] Beiter M.: Magnitude of the coefficients of the cyclotomic polynomial F pqr . II. Duke Math. J. 38, 591–594 (1971) · Zbl 0221.10018
[21] Boca F.P.: Products of matrices $${\(\backslash\)left[\(\backslash\)begin{array}{ll}1\(\backslash\)quad 1\(\backslash\)\(\backslash\) 0\(\backslash\)quad1\(\backslash\)end{array} \(\backslash\)right]}$$ and $${\(\backslash\)left[\(\backslash\)begin{array}{ll}1\(\backslash\)quad 0\(\backslash\)\(\backslash\) 1\(\backslash\)quad1\(\backslash\)end{array} \(\backslash\)right]}$$ and the distribution of reduced quadratic irrationals. J. Reine Angew. Math. 606, 149–165 (2007)
[22] Boca F.P.: Distribution of angles between geodesic rays associated with hyperbolic lattice points. Q. J. Math. 58, 281–295 (2007) · Zbl 1197.11134
[23] Boca F.P., Cobeli C., Zaharescu A.: Distribution of lattice points visible from the origin. Comm. Math. Phys. 213, 433–470 (2000) · Zbl 0989.11049
[24] Boca F.P., Cobeli C., Zaharescu A.: A conjecture of R.R. Hall on Farey points. J. Reine Angew. Math. 535, 207–236 (2001) · Zbl 1006.11053
[25] Boca F.P., Gologan R.N., Zaharescu A.: The statistics of the trajectory of a certain billiard in a flat two-torus. Comm. Math. Phys. 240, 53–73 (2003) · Zbl 1078.37006
[26] F.P. Boca and A. Zaharescu, Farey fractions and two-dimensional tori, In: Noncommutative Geometry and Number Theory, Aspects Math., E37, Vieweg, Wiesbaden, 2006, pp. 57–77. · Zbl 1106.11005
[27] Boca F.P., Zaharescu A.: On the correlations of directions in the Euclidean plane. Trans. Amer. Math. Soc. 358, 1797–1825 (2006) · Zbl 1154.11022
[28] Boca F.P., Zaharescu A.: The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit. Comm. Math. Phys. 269, 425–471 (2007) · Zbl 1143.37002
[29] Bombieri E.: On exponential sums in finite fields. Amer. J. Math. 88, 71–105 (1966) · Zbl 0171.41504
[30] Bombieri E., Friedlander J.B., Iwaniec H.: Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2, 215–224 (1989) · Zbl 0674.10036
[31] Bourgain J.: Mordell’s exponential sum estimate revisited. J. Amer. Math. Soc. 18, 477–499 (2005) · Zbl 1072.11063
[32] Bourgain J.: More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1, 1–32 (2005) · Zbl 1173.11310
[33] J. Bourgain, New encounters in combinatorial number theory: From the Kakeya problem to cryptography, In: Perspectives in Analysis, Math. Phys. Stud., 27, Springer-Verlag, 2005, pp. 17–26. · Zbl 1184.11003
[34] Bourgain J.: Estimates of polynomial exponential sums. Israel J. Math. 176, 221–240 (2010) · Zbl 1205.11094
[35] Bourgain J., Cochrane T., Paulhus J., Pinner C.: Decimations of l-sequences and permutations of even residues mod p. SIAM J. Discrete Math. 23, 842–857 (2009) · Zbl 1213.11006
[36] Bourgain J., Cochrane T., Paulhus J., Pinner C.: On the parity of k-th powers modulo p. A generalization of a problem of Lehmer. Acta Arith. 147, 173–203 (2011) · Zbl 1239.11106
[37] J. Bourgain, M.Z. Garaev, S.V. Konyagin and I.E. Shparlinski, On the hidden shifted power problem, SIAM J. Comput., to appear. · Zbl 1311.11111
[38] J. Bourgain, M.Z. Garaev, S.V. Konyagin and I.E. Shparlinski, On congruences with products of variables from short intervals and applications, Proc. Steklov Inst. Math., to appear. · Zbl 1301.11041
[39] J. Bourgain, M.Z. Garaev, S.V. Konyagin and I.E. Shparlinski, Multiplicative congruences with variables from short intervals, preprint, 2012, arXiv:1210.6429. · Zbl 1385.11002
[40] J. Bourgain and Ya.G. Sinai, Limiting behavior of large Frobenius numbers, Uspekhi Mat. Nauk, 62 (2007), no. 4, 77–90 (in Russian); transl. as Russian Math. Surveys. · Zbl 1151.11046
[41] P. Brass, W. Moser and J. Pach, Research Problems in Discrete Geometry, Springer-Verlag, 2005.
[42] T. Browning and A. Haynes, Incomplete Kloosterman sums and multiplicative inverses in short intervals, preprint, 2012, arXiv:1204.6374. · Zbl 1271.11082
[43] V.A. Bykovskiĭ, Asymptotic properties of lattice points (a 1,a 2) that satisfy the congruence $${a_1a_2\(\backslash\)equiv l\(\backslash\);(\(\backslash\)mod q)}$$ , In: Analytic Number Theory and the Theory of Functions. 4, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 112, Nauka Leningrad. Otdel., Leningrad, 1981, pp. 5–25 (in Russian).
[44] Bykovskiĭ V.A.: An estimate for the dispersion of lengths of finite continued fractions. J. Math. Sci. (N. Y.) 146, 5634–5643 (2007) · Zbl 1204.11016
[45] V.A. Bykovskiĭ and A.V. Ustinov, The statistics of particle trajectories in the homogeneous Sinai problem for a two-dimensional lattice, Funktsional. Anal. i Prilozhen, 42 (2008), no. 3, 10–22 (in Russian); transl. as Funct. Anal. Appl.
[46] V.A. Bykovskiĭ and A.V. Ustinov, The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice, Izv. Ross. Akad. Nauk Ser. Mat., 73 (2009), no. 4, 17–36 (in Russian); transl. as Izv. Math.
[47] T.H. Chan, Distribution of difference between inverses of consecutive integers modulo p, Integers, 4 (2004), A03, 11 pp. · Zbl 1083.11063
[48] Chan T.H.: Approximating reals by sums of two rationals. J. Number Theory 128, 1182–1194 (2008) · Zbl 1141.11037
[49] Chan T.H.: Approximating reals by sums of rationals. J. Number Theory 129, 316–324 (2009) · Zbl 1228.11097
[50] Chan T.H.: A short note on the difference between inverses of consecutive integers modulo p. Integers 9, 699–702 (2009) · Zbl 1223.11117
[51] Chan T.H.: An almost all result on $${q_1q_2 \(\backslash\)equiv c\(\backslash\);(\(\backslash\)mod q)}$$ . Monatsh. Math. 162, 29–39 (2011) · Zbl 1264.11086
[52] Chan T.H., Shparlinski I.E.: On the concentration of points on modular hyperbolas and exponential curves. Acta Arith. 142, 59–66 (2010) · Zbl 1198.11002
[53] Cilleruelo J., Garaev M.Z.: Concentration of points on two and three dimensional modular hyperbolas and applications. Geom. Funct. Anal. 21, 892–904 (2011) · Zbl 1225.11004
[54] J. Cilleruelo, M.Z. Garaev, A. Ostafe and I.E. Shparlinski, On the concentration of points of polynomial maps and applications, Math. Z., to appear. · Zbl 1285.11027
[55] Cobeli C., Gonek S.M., Zaharescu A.: The distribution of patterns of inverses modulo a prime. J. Number Theory 101, 209–222 (2003) · Zbl 1086.11045
[56] Cobeli C., Vâjâitu M., Zaharescu A.: Average estimates for the number of tuples of inverses mod p in short intervals. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 43, 155–164 (2000)
[57] Cobeli C., Vâjâitu M., Zaharescu A.: Distribution of gaps between the inverses mod q. Proc. Edinb. Math. Soc. 46(2), 185–203 (2003) · Zbl 1034.11055
[58] Cobeli C., Zaharescu A.: The order of inverses mod q. Mathematika 47, 87–108 (2000) · Zbl 1031.11058
[59] Cobeli C., Zaharescu A.: Generalization of a problem of Lehmer. Manuscripta Math. 104, 301–307 (2001) · Zbl 1034.11042
[60] Cobeli C., Zaharescu A.: On the distribution of the $${\(\backslash\)mathbb{F}_p}$$ -points on an affine curve in r dimensions. Acta Arith. 99, 321–329 (2001) · Zbl 1025.11021
[61] Cochrane T., Pinner C., Rosenhouse J.: Sparse polynomial exponential sums. Acta Arith. 108, 37–52 (2003) · Zbl 1158.11332
[62] Cochrane T., Sih S.: The congruence $${x_1x_2 \(\backslash\)equiv x_3x_4\(\backslash\);(\(\backslash\)mod m)}$$ and mean values of character sums. J. Number Theory 130, 767–785 (2010) · Zbl 1217.11002
[63] Cochrane T., Zheng Z.: High order moments of character sums. Proc. Amer. Math. Soc. 126, 951–956 (1998) · Zbl 0893.11033
[64] Cojocaru A.C., Hall C.: Uniform results for Serre’s theorem for elliptic curves. Int. Math. Res. Not. 2005, 3065–3080 (2005) · Zbl 1178.11045
[65] Cojocaru A.C., Shparlinski I.E.: Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average. Proc. Amer. Math. Soc. 136, 1977–1986 (2008) · Zbl 1154.11007
[66] E.I. Dinaburg and Ya.G. Sinai, The statistics of the solutions of the integer equation axy = {\(\pm\)}1, Funktsional. Anal. i Prilozhen., 24 (1990), no. 3, 1–8 (in Russian); transl. as Funct. Anal. Appl.
[67] D. Dolgopyat, On the distribution of the minimal solution of a linear Diophantine equation with random coefficients, Funktsional. Anal. i Prilozhen., 28 (1994), no. 3, 22–34 (in Russian); transl. as Funct. Anal. Appl. · Zbl 0824.11046
[68] M. Drmota and R. Tichy, Sequences, Discrepancies and Applications, Springer-Verlag, 1997. · Zbl 0877.11043
[69] Duke W., Friedlander J.B., Iwaniec H.: Bilinear forms with Kloosterman fractions. Invent. Math. 128, 23–43 (1997) · Zbl 0873.11050
[70] Duke W., Rudnick Z., Sarnak P.: Density of integer points on affine homogeneous varieties. Duke Math. J. 71, 143–179 (1993) · Zbl 0798.11024
[71] Dvir Z.: On the size of Kakeya sets in finite fields. J. Amer. Math. Soc. 22, 1093–1097 (2009) · Zbl 1202.52021
[72] D. Eichhorn, M.R. Khan, A.H. Stein and C.L. Yankov, Sums and differences of the coordinates of points on modular hyperbolas, In: Combinatorial Number Theory, Walter de Gruyter, 2009, pp. 17–39. · Zbl 1178.11004
[73] Erdos P., Odlyzko A.M., Sárközy A.: On the residues of products of prime numbers. Period. Math. Hungar 18, 229–239 (1987) · Zbl 0625.10035
[74] Ferguson R., Hoffman C., Luca F., Ostafe A., Shparlinski I.E.: Some additive combinatorics problems in matrix rings. Rev. Mat. Complut. 23, 501–513 (2010) · Zbl 1208.11037
[75] Ford K.: The distribution of integers with a divisor in a given interval. Ann. of Math. 168(2), 367–433 (2008) · Zbl 1181.11058
[76] Ford K., Khan M.R., Shparlinski I.E.: Geometric properties of points on modular hyperbolas. Proc. Amer. Math. Soc. 138, 4177–4185 (2010) · Zbl 1204.11004
[77] Ford K., Khan M.R., Shparlinski I.E., Yankov C.L.: On the maximal difference between an element and its inverse in residue rings. Proc. Amer. Math. Soc. 133, 3463–3468 (2005) · Zbl 1131.11005
[78] Fouvry É.: Sur le problème des diviseurs de Titchmarsh. J. Reine Angew. Math. 357, 51–76 (1985) · Zbl 0547.10039
[79] Fouvry É.: Consequences of a result of N. Katz and G. Laumon concerning trigonometric sums. Israel J. Math. 120, 81–96 (2000) · Zbl 1010.11045
[80] Fouvry É., Katz N.M.: A general stratification theorem for exponential sums, and applications. J. Reine Angew. Math. 540, 115–166 (2001) · Zbl 0986.11054
[81] Fouvry É., Michel P.: Sur certaines sommes d’exponentielles sur les nombres premiers. Ann. Sci. École Norm. Sup. 31(4), 93–130 (1998) · Zbl 0915.11045
[82] Fouvry É., Shparlinski I.E.: On a ternary quadratic form over primes. Acta Arith. 150, 285–314 (2011) · Zbl 1243.11093
[83] Fouvry É., Shparlinski I.E.: Smooth shifted monomial products. Publ. Math. Debrecen 79, 423–432 (2011) · Zbl 1249.11095
[84] Friedlander J.B., Iwaniec H.: Incomplete Kloosterman sums and a divisor problem. Ann. of Math. 121(2), 319–350 (1985) · Zbl 0572.10029
[85] Friedlander J.B., Iwaniec H.: The divisor problem for arithmetic progressions. Acta Arith. 45, 273–277 (1985) · Zbl 0572.10033
[86] Friedlander J.B., Kurlberg P., Shparlinski I.E.: Products in residue classes. Math. Res. Lett. 15, 1133–1147 (2008) · Zbl 1182.11046
[87] Friedlander J.B., Luca F.: Residue classes having tardy totients. Bull. Lond. Math. Soc. 40, 1007–1016 (2008) · Zbl 1176.11045
[88] Friedlander J.B., Shparlinski I.E.: Least totient in a residue class. Bull. Lond. Math. Soc. 39, 425–432 (2007) · Zbl 1127.11056
[89] Fujii A.: On a problem of Dinaburg and Sinai. Proc. Japan Acad. Ser. A Math. Sci. 68, 198–203 (1992) · Zbl 0779.11032
[90] Fujii A., Kitaoka Y.: On plain lattice points whose coordinates are reciprocals modulo a prime. Nagoya Math. J. 147, 137–146 (1997) · Zbl 0917.11051
[91] Fukshansky L., Robins S.: Frobenius problem and the covering radius of a lattice. Discrete Comput. Geom. 37, 471–483 (2007) · Zbl 1136.11307
[92] Gallot Y., Moree P.: Ternary cyclotomic polynomials having a large coefficient. J. Reine Angew. Math. 632, 105–125 (2009) · Zbl 1230.11030
[93] Garaev M.Z.: Character sums in short intervals and the multiplication table modulo a large prime. Monatsh. Math. 148, 127–138 (2006) · Zbl 1142.11062
[94] Garaev M.Z.: On the logarithmic factor in error term estimates in certain additive congruence problems. Acta Arith. 124, 27–39 (2006) · Zbl 1158.11002
[95] Garaev M.Z.: A note on the least totient of a residue class. Q. J. Math. 60, 53–56 (2009) · Zbl 1247.11126
[96] Garaev M.Z.: Estimation of Kloosterman sums with primes and its application. Mat. Zametki 88, 365–373 (2010) (in Russian) · Zbl 1268.11108
[97] Garaev M.Z.: On multiplicative congruences. Math. Z. 272, 473–482 (2012) · Zbl 1259.11075
[98] Garaev M.Z., Garcia V.C.: The equation x 1 x 2 = x 3 x 4 + {\(\lambda\)} in fields of prime order and applications. J. Number Theory 128, 2520–2537 (2008) · Zbl 1225.11005
[99] Garaev M.Z., Karatsuba A.A.: On character sums and the exceptional set of a congruence problem. J. Number Theory 114, 182–192 (2005) · Zbl 1135.11044
[100] Garaev M.Z., Karatsuba A.A.: The representation of residue classes by products of small integers. Proc. Edinb. Math. Soc. 50(2), 363–375 (2007) · Zbl 1197.11003
[101] Garaev M.Z., Kueh K.-L.: Distribution of special sequences modulo a large prime. Int. J. Math. Math. Sci. 2003, 3189–3194 (2003) · Zbl 1037.11002
[102] Gonek S.M., Krishnaswami G.S., Sondhi V. L.: The distribution of inverses modulo a prime in short intervals. Acta Arith. 102, 315–322 (2002) · Zbl 0994.11034
[103] Goresky M., Klapper A.: Arithmetic crosscorrelations of feedback with carry shift register sequences. IEEE Trans. Inform. Theory 43, 1342–1345 (1997) · Zbl 0878.94047
[104] Goresky M., Klapper A., Murty R., Shparlinski I.E.: On decimations of -sequences. SIAM J. Discrete Math. 18, 130–140 (2004) · Zbl 1114.11015
[105] Granville A., Shparlinski I.E., Zaharescu A.: On the distribution of rational functions along a curve over $${\(\backslash\)mathbb{F}_p}$$ and residue races. J. Number Theory 112, 216–237 (2005) · Zbl 1068.11043
[106] L. Guth and N.H. Katz, On the Erdos distinct distance problem in the plane, preprint, 2011, arXiv:1011.4105.
[107] R.K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994. · Zbl 0805.11001
[108] R.R. Hall and G. Tenenbaum, Divisors, Cambridge Tracts in Math., 90, Cambridge Univ. Press, 1988.
[109] Hanrahan S., Khan M.R.: The cardinality of the value sets modulo n of x 2 + x and x 2 + y 2. Involve 3, 171–182 (2010) · Zbl 1218.11004
[110] Heath-Brown D.R.: The divisor function d 3(n) in arithmetic progressions. Acta Arith. 47, 29–56 (1986) · Zbl 0549.10034
[111] Heath-Brown D.R.: Pair correlation for fractional parts of {\(\alpha\)} n 2. Math. Proc. Cambridge Philos. Soc. 148, 385–407 (2010) · Zbl 1239.11081
[112] Hooley C.: An asymptotic formula in the theory of numbers. Proc. London Math. Soc. 7(3), 396–413 (1957) · Zbl 0079.27301
[113] Hooley C.: On the greatest prime factor of a cubic polynomial. J. Reine Angew. Math. 303/304, 21–50 (1978) · Zbl 0391.10028
[114] S. Hu and Y. Li, On a uniformly distributed phenomenon in matrix groups, preprint, 2011, arXiv:1103.3928.
[115] M.N. Huxley, Large values of Dirichlet polynomials. III, Acta Arith., 26 (1974/1975), 435–444. · Zbl 0268.10026
[116] Huxley M.N., Jutila M.: Large values of Dirichlet polynomials. IV. Acta Arith. 32, 297–312 (1977) · Zbl 0352.10019
[117] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., 53, Amer. Math. Soc., Providence, RI, 2004. · Zbl 1059.11001
[118] Jutila M.: Zero-density estimates for L-functions. Acta Arith. 32, 55–62 (1977) · Zbl 0307.10045
[119] Karatsuba A.A.: Sums of characters with prime numbers. Izv. Akad. Nauk SSSR Ser. Mat. 34, 299–321 (1970) (in Russian)
[120] A.A. Karatsuba, Fractional parts of functions of a special form, Izv. Ross. Akad. Nauk Ser. Mat., 59 (1995), no. 4, 61–80 (in Russian); transl. as Izv. Math. · Zbl 0874.11050
[121] A.A. Karatsuba, Analogues of Kloosterman sums, Izv. Ross. Akad. Nauk Ser. Mat., 59 (1995), no. 5, 93–102 (in Russian); transl. as Izv. Math. · Zbl 0897.11028
[122] N. Katz and T. Tao, Recent progress on the Kakeya conjecture, In: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, Publ. Mat., Extra, Univ. Autònoma Barcelona, 2002, pp. 161–180.
[123] Khan M.R.: An optimization with a modular constraint: 10736. Amer. Math. Monthly 108, 374–375 (2001)
[124] Khan M.R.: Modular hyperbolas and the coefficients of (x + 6 + x) k . Integers 11, 469–476 (2011) · Zbl 1253.11023
[125] Khan M.R., Shparlinski I.E.: On the maximal difference between an element and its inverse modulo n. Period. Math. Hungar. 47, 111–117 (2003) · Zbl 1047.11003
[126] Khan M.R., Shparlinski I.E., Yankov C.L.: On the convex closure of the graph of modular inversions. Experiment. Math. 17, 91–104 (2008) · Zbl 1234.11005
[127] Konyagin S.V., Shparlinski I.E.: On convex hull of points on modular hyperbolas. Moscow J. Comb. Number Theory 1, 43–51 (2011) · Zbl 1280.11003
[128] M.A. Korolev, Incomplete Kloosterman sums and their applications, Izv. Ross. Akad. Nauk Ser. Mat., 64 (2000), no. 6, 41–64 (in Russian); transl. as Izv. Math. · Zbl 1030.11039
[129] N.V. Kuznetsov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums, Mat. Sb. (N.S.), 111 (1980), 334–383 (in Russian); transl. as Math. USSR-Sb. · Zbl 0427.10016
[130] Laczkovich M.: Discrepancy estimates for sets with small boundary. Studia Sci. Math. Hungar. 30, 105–109 (1995) · Zbl 0851.11045
[131] Le Boudec P.: Manin’s conjecture for two quartic del Pezzo surfaces with 3A 1 and A 1 + A 2 singularity types. Acta Arith. 151, 109–163 (2012) · Zbl 1248.11045
[132] Le Boudec P.: Power-free values of the polynomial t 1...t r 1. Bull. Aust. Math. Soc. 85, 154–163 (2012) · Zbl 1292.11112
[133] P. Le Boudec, Manin’s conjecture for a cubic surface with 2A 2 + A 1 singularity type, Proc. Cambridge Philos. Soc., to appear. · Zbl 1253.14022
[134] Liu H.N.: A note on Lehmer k-tuples. Int. J. Number Theory 5, 1169–1178 (2009) · Zbl 1229.11109
[135] Liu H.N., Zhang W.: On a problem of D.H. Lehmer, Acta Math. Sin. (Engl. Ser.) 22, 61–68 (2006) · Zbl 1100.11029
[136] Liu H.N., Zhang W.: Hybrid mean value on the difference between a quadratic residue and its inverse modulo p. Publ. Math. Debrecen 69, 227–243 (2006) · Zbl 1174.11385
[137] Liu H.N., Zhang W.: General Kloosterman sums and the difference between an integer and its inverse modulo q. Acta Math. Sin. (Engl. Ser.) 23, 77–82 (2007) · Zbl 1106.11031
[138] Liu H.N., Zhang W.: Mean value on the difference between a quadratic residue and its inverse modulo p. Acta Math. Sin. (Engl. Ser.) 23, 915–924 (2007) · Zbl 1234.11105
[139] Liu H.N., Zhang W.: Hybrid mean value results for a generalization on a problem of D.H. Lehmer and hyper-Kloosterman sums. Osaka J. Math. 44, 615–637 (2007) · Zbl 1147.11043
[140] Liu H.N., Zhang W.: Hybrid mean value on the difference between an integer and its inverse modulo q. Ark. Mat. 46, 337–347 (2008) · Zbl 1229.11108
[141] Louboutin S.R., Rivat J., Sárközy A.: On a problem of D.H. Lehmer. Proc. Amer. Math. Soc. 135, 969–975 (2007) · Zbl 1160.11039
[142] Lu Y., Yi Y.: On the generalization of the D.H. Lehmer problem. Acta Math. Sin. (Engl. Ser.) 25, 1269–1274 (2009) · Zbl 1170.11022
[143] Lu Y., Yi Y.: On the generalization of the D.H. Lehmer problem. II. Acta Arith. 142, 179–186 (2010) · Zbl 1194.11096
[144] Lu Y., Yi Y.: Partitions involving D.H. Lehmer numbers. Monatsh. Math. 159, 45–58 (2010) · Zbl 1244.11086
[145] Lu Y., Yi Y.: A note on the Lehmer problem over short intervals. Acta Math. Sin. (Engl. Ser.) 27, 1115–1120 (2011) · Zbl 1278.11078
[146] Luo W.: Bounds for incomplete hyper-Kloosterman sums. J. Number Theory 75, 41–46 (1999) · Zbl 0923.11118
[147] K.-H. Mak and A. Zaharescu, Lehmer points and visible points on affine varieties over finite fields, preprint, 2011, arXiv:1110.4691.
[148] Marklof J., Strömbergsson A.: Equidistribution of Kronecker sequences along closed horocycles. Geom. Funct. Anal. 13, 1239–1280 (2003) · Zbl 1048.37009
[149] Matomäki K.: On the greatest prime factor of ab + 1. Acta Math. Hungar. 124, 115–123 (2009) · Zbl 1212.11080
[150] Merel L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124, 437–449 (1996) · Zbl 0936.11037
[151] N.G. Moshchevitin, On numbers with missing digits: Solvability of the congruences $${x_1x_2 \(\backslash\)equiv \(\backslash\)lambda\(\backslash\);(\(\backslash\)mod p)}$$ , Dokl. Akad. Nauk, 410 (2006), 730–733 (in Russian); transl. as Dokl. Math.
[152] N.G. Moshchevitin, Sets of the form $${\(\backslash\)fancyscript{A}+\(\backslash\)fancyscript{B}}$$ and finite continued fractions, Mat. Sb., 198 (2007), no. 4, 95–116 (in Russian); transl. as Sb. Math.
[153] N.G. Moshchevitin and I.D. Shkredov, On the multiplicative properties modulo m of numbers with missing digits, Mat. Zametki, 81 (2007), 385–404 (in Russian); transl. as Math. Notes. · Zbl 1198.11006
[154] Niederreiter H., Wills J.M.: Diskrepanz und Distanz von Maßen bezüglich konvexer und Jordanscher Mengen. Math. Z. 144, 125–134 (1975) · Zbl 0295.28028
[155] Petrov F.V.: Estimates for the number of rational points on convex curves and surfaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 344, 174–189 (2007) (in Russian)
[156] Rakhmonov Z.Kh.: On the distribution of the values of Dirichlet characters and their applications. Proc. Steklov Inst. Math. 207, 263–272 (1995)
[157] J.L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Lecture Ser. Math. Appl., 30, Oxford Univ. Press, Oxford, 2005.
[158] Rényi A., Sulanke R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 75–84 (1963) · Zbl 0118.13701
[159] Rieger G.J.: Über die Gleichung adc = 1 und Gleichverteilung. Math. Nachr. 162, 139–143 (1993) · Zbl 0820.11013
[160] Roettger C.: Counting invertible matrices and uniform distribution. J. Théor. Nombres Bordeaux 17, 301–322 (2005) · Zbl 1101.11011
[161] M. Rubinstein, Hide and seek–a naive factoring algorithm, preprint, 2006, arXiv: math/0610612.
[162] Rudnick Z., Sarnak P., Zaharescu A.: The distribution of spacings between the fractional parts of n 2{\(\alpha\)}. Invent. Math. 145, 37–57 (2001) · Zbl 1006.11041
[163] Ruzsa I.Z., Schinzel A.: An application of Kloosterman sums. Compositio Math. 96, 323–330 (1995) · Zbl 0833.11039
[164] Saias É.: Entiers à diviseurs denses. I. J. Number Theory 62, 163–191 (1997) · Zbl 0872.11039
[165] J.-C. Schlage-Puchta, An estimate for Frobenius’ Diophantine problem in three dimensions, J. Integer Seq., 8 (2005), 05.1.7, 4 pp., http://www.cs.uwaterloo.ca/journals/JIS/vol8.html . · Zbl 1068.11018
[166] I.A. Semaev, On the number of small solutions of a linear homogeneous congruence, Mat. Zametki, 50 (1991), no. 4, 102–107 (in Russian);transl. as Math. Notes. · Zbl 0736.11002
[167] Semaev I.A.: An algorithm for evaluation of discrete logarithms in some nonprime finite fields. Math. Comp. 67, 1679–1689 (1998) · Zbl 0922.11106
[168] ShchurV. Sinai Ya.G., Ustinov A.V.: Limiting distribution of Frobenius numbers for n = 3. J. Number Theory 129, 2778–2789 (2009) · Zbl 1229.11050
[169] V. Shelestunova, Upper bounds for the number of integral points on quadratic curves and surfaces, Ph. D. thesis, Univ. of Waterloo, Ontario, Canada, 2010.
[170] Shparlinski I.E.: On exponential sums with sparse polynomials and rational functions. J. Number Theory 60, 233–244 (1996) · Zbl 0881.11081
[171] Shparlinski I.E.: Primitive points on modular hyperbola. Bull. Pol. Acad. Sci. Math. 54, 193–200 (2006) · Zbl 1153.11322
[172] Shparlinski I.E.: On the distribution of points on multidimensional modular hyperbolas. Proc. Japan Acad. Ser. A Math. Sci. 83, 5–9 (2007) · Zbl 1123.11026
[173] Shparlinski I.E.: Bounds of incomplete multiple Kloosterman sums. J. Number Theory 126, 68–73 (2007) · Zbl 1143.11026
[174] Shparlinski I.E.: Distribution of modular inverses and multiples of small integers and the Sato–Tate conjecture on average. Michigan Math. J. 56, 99–111 (2008) · Zbl 1225.11101
[175] Shparlinski I.E.: On the Euler function on differences between the coordinates of points on modular hyperbolas. Bull. Pol. Acad. Sci. Math. 56, 1–7 (2008) · Zbl 1144.11004
[176] Shparlinski I.E.: Approximation by several rationals. Bull. Aust. Math. Soc. 77, 325–329 (2008) · Zbl 1157.11029
[177] I.E. Shparlinski, On a generalised Lehmer problem for arbitrary powers, In: Contributions in General Algebra. II, East-West J. Math., Special, Khon Kaen Univ., Bangkok, 2008, pp. 197–216. · Zbl 1182.11005
[178] Shparlinski I.E.: On some weighted average values of L-functions. Bull. Aust. Math. Soc. 79, 183–186 (2009) · Zbl 1210.11093
[179] Shparlinski I.E.: On a generalisation of a Lehmer problem. Math. Z. 263, 619–631 (2009) · Zbl 1269.11068
[180] Shparlinski I.E.: On the distribution of solutions to linear equations. Glas. Math. Ser. III 44, 7–10 (2009) · Zbl 1234.11031
[181] Shparlinski I.E.: On small solutions to quadratic congruences. J. Number Theory 131, 1105–1111 (2011) · Zbl 1228.11044
[182] Shparlinski I.E.: On the restricted divisor function in arithmetic progressions. Rev. Mat. Iberoam. 28, 231–238 (2012) · Zbl 1323.11073
[183] I.E. Shparlinski, On products of primes and almost primes in arithmetic progressions, Period. Math. Hungar., to appear. · Zbl 1299.11067
[184] Shparlinski I.E., Voloch J.F.: Visible points on curves over finite fields. Bull. Pol. Acad. Sci. Math. 55, 193–199 (2007) · Zbl 1156.11006
[185] Shparlinski I.E., Winterhof A.: On the number of distances between the coordinates of points on modular hyperbolas. J. Number Theory 128, 1224–1230 (2008) · Zbl 1137.11063
[186] Shparlinski I.E., Winterhof A.: Visible points on multidimensional modular hyperbolas. J. Number Theory 128, 2695–2703 (2008) · Zbl 1204.11006
[187] Shparlinski I.E., Winterhof A.: Partitions into two Lehmer numbers. Monatsh. Math. 160, 429–441 (2010) · Zbl 1223.11119
[188] Stewart C.L.: On the greatest prime factor of integers of the form ab + 1. Period. Math. Hungar. 43, 81–91 (2001) · Zbl 0999.11055
[189] Šunić Z.: Frobenius problem and dead ends in integers. J. Number Theory 128, 1211–1223 (2008) · Zbl 1165.11030
[190] Taylor R.: Automorphy for some l-adic lifts of automorphic mod l Galois representations. II. Publ. Math. Inst. Hautes Études Sci. 108, 183–239 (2008) · Zbl 1169.11021
[191] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Stud. Adv. Math., 46, Cambridge Univ. Press, Cambridge, 1995.
[192] Truelsen J.L.: Divisor problems and the pair correlation for the fractional parts of n 2{\(\alpha\)}. Int. Math. Res. Not. IMRN 2010, 3144–3183 (2010) · Zbl 1211.11087
[193] Ustinov A.V.: On the statistical properties of finite continued fractions. J. Math. Sci. (N. Y.) 137, 4722–4738 (2006) · Zbl 1072.11060
[194] A.V Ustinov., Calculation of variance in a problem in the theory of continued fractions, Mat. Sb., 198 (2007), no. 6, 139–158 (in Russian); transl. as Sb. Math. · Zbl 1197.11096
[195] A.V. Ustinov, Asymptotic behavior of the first and second moments for the number of steps in the Euclid algorithm, Izv. Ross. Akad. Nauk Ser. Mat., 72 (2008), no. 5, 189–224 (in Russian); transl. as Izv. Math.
[196] A.V. Ustinov, On the number of solutions of the congruence $${xy \(\backslash\)equiv l\(\backslash\);(\(\backslash\)mod q)}$$ under the graph of a twice continuously differentiable function, Algebra i Analiz, 20 (2008), no. 5, 186–216 (in Russian).
[197] A.V. Ustinov, Solution of the Arnol’d problem on weak asymptotics for Frobenius numbers with three arguments, Mat. Sb., 200 (2009), no. 4, 131–160 (in Russian); transl. as Sb. Math.
[198] Ustinov A.V.: On the distribution of integer points. Dal’nevost. Mat. Zh. 9, 176–181 (2009) (in Russian)
[199] Ustinov A.V.: The mean number of steps in the Euclidean algorithm with least absolute-value remainders. Mat. Zametki 85, 153–156 (2009) (in Russian);transl. as Math. Notes. · Zbl 1208.11014
[200] Ustinov A.V.: On the statistical properties of elements of continued fractions. Dokl. Akad. Nauk 424, 459–461 (2009) (in Russian); transl. as Dokl. Math. · Zbl 1273.11016
[201] A.V. Ustinov, On the distribution of Frobenius numbers with three arguments, Izv. Ross. Akad. Nauk Ser. Mat., 74 (2010), no. 5, 145–170 (in Russian); transl. as Izv. Math.
[202] Vâjâitu M., Zaharescu A.: Distribution of values of rational maps on the $${\(\backslash\)mathbb{F}_p}$$ -points on an affine curve. Monatsh. Math. 136, 81–86 (2002) · Zbl 1029.11022
[203] Vaughan R.C., Wooley T.D.: Further improvements in Waring’s problem. Acta Math. 174, 147–240 (1995) · Zbl 0849.11075
[204] Vinh L.A.: Distribution of determinant of matrices with restricted entries over finite fields. J. Comb. Number Theory 1, 203–212 (2009) · Zbl 1234.11030
[205] L.A. Vinh, On the distribution of permanents of matrices over finite fields, In: European Conference on Combinatorics, Graph Theory and Applications, Electron. Notes Discrete. Math., 34, Elsevier Sci. B. V., Amsterdam, 2009, pp. 519–523. · Zbl 1273.05035
[206] Wang Y., Li H.: On s-dimensional incomplete Kloosterman sums. J. Number Theory 130, 1602–1608 (2010) · Zbl 1208.11096
[207] Weili Y.: On the generalization of the D.H. Lehmer problem and its mean value. JP J. Algebra Number Theory Appl. 6, 479–491 (2006) · Zbl 1127.11068
[208] Weyl H.: On the volume of tubes, Amer. J. Math. 61, 461–472 (1939) · JFM 65.0796.01
[209] Xi P., Yi Y.: Generalized D.H. Lehmer problem over short intervals. Glasg. Math. J. 53, 293–299 (2011) · Zbl 1273.11008
[210] Xiong M., Zaharescu A.: Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219, 523–553 (2008) · Zbl 1194.11066
[211] Xu Z.: D.H. Lehmer problem over half intervals. J. Korean Math. Soc. 46, 493–511 (2009) · Zbl 1193.11091
[212] Xu Z., Zhang W.: On a problem of D.H. Lehmer over short intervals. J. Math. Anal. Appl. 320, 756–770 (2006) · Zbl 1098.11050
[213] Xu Z., Zhang W.: A problem of D.H. Lehmer and its mean value. Math. Nachr. 281, 596–606 (2008) · Zbl 1153.11050
[214] Yi Y., Zhang W.: On the generalization of a problem of D.H. Lehmer. Kyushu J. Math. 56, 235–241 (2002) · Zbl 1136.11323
[215] Yu G.: Rank 0 quadratic twists of a family of elliptic curves. Compositio Math. 135, 331–356 (2003) · Zbl 1090.11038
[216] Yuan Y., Yiwei H.: On a generalization of D.H. Lehmer problem. JP J. Algebra Number Theory Appl. 14, 37–50 (2009) · Zbl 1246.11163
[217] Zaharescu A.: The distribution of the values of a rational function modulo a big prime. J. Théor. Nombres Bordeaux 15, 863–872 (2003) · Zbl 1093.11062
[218] Zhang T., Xue X.: On the r-th hyper-Kloosterman sums and a problem of D.H. Lehmer. J. Korean Math. Soc. 46, 733–746 (2009) · Zbl 1179.11023
[219] Zhang T., Zhang W.: A generalization on the difference between an integer and its inverse modulo q. II. Proc. Japan Acad. Ser. A Math. Sci. 81, 7–11 (2005) · Zbl 1088.11072
[220] Zhang W.: On a problem of D.H. Lehmer and its generalization. Compositio Math. 86, 307–316 (1993) · Zbl 0783.11002
[221] Zhang W.: On a problem of D.H. Lehmer and its generalization II. Compositio Math. 91, 47–56 (1994) · Zbl 0798.11001
[222] Zhang W.: On the difference between a D.H. Lehmer number and its inverse modulo q. Acta Arith. 68, 255–263 (1994) · Zbl 0826.11003
[223] Zhang W.: On the difference between an integer and its inverse modulo n. J. Number Theory 52, 1–6 (1995) · Zbl 0826.11002
[224] Zhang W.: On the distribution of inverses modulo n. J. Number Theory 61, 301–310 (1996) · Zbl 0874.11006
[225] Zhang W.: On a problem of P. Gallagher. Acta Math. Hungar. 78, 345–357 (1998) · Zbl 0903.11022
[226] Zhang W.: On the distribution of inverses modulo p. II. Acta Arith. 100, 189–194 (2001) · Zbl 0997.11077
[227] Zhang W.: On a problem of D.H. Lehmer and Kloosterman sums. Monatsh. Math. 139, 247–257 (2003) · Zbl 1094.11028
[228] Zhang W.: On the mean value of L-functions with the weight of character sums. J. Number Theory 128, 2459–2466 (2008) · Zbl 1181.11054
[229] Zhang W., Xu Z., Yi Y.: A problem of D.H. Lehmer and its mean square value formula. J. Number Theory 103, 197–213 (2003) · Zbl 1046.11070
[230] Zhang W., Yi Y.: Some applications of Bombieri’s estimate for exponential sums. Acta Arith. 107, 245–250 (2003) · Zbl 1126.11339
[231] Zheng Z.: The distribution of zeros of an irreducible curve over a finite field. J. Number Theory 59, 106–118 (1996) · Zbl 0862.11041
[232] Zheng Z., Cochrane T.: Distribution of primitive {\(\lambda\)}-roots of composite moduli. II. Chinese Ann. Math. Ser. B 27, 549–552 (2006) · Zbl 1117.11043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.