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Uspekhi Mat. Nauk, 2007, Volume 62, Issue 4(376), Pages 77–90 (Mi umn7150)  

This article is cited in 17 scientific papers (total in 18 papers)

Limit behaviour of large Frobenius numbers

J. Bourgaina, Ya. G. Sinaib

a Institute for Advanced Study, School of Mathematics
b Princeton University, Department of Mathematics

Abstract: This is an investigation of the problem of the asymptotic distribution of the Frobenius numbers of $n$ relatively prime integers. For $n=3$ virtually definitive results are obtained. For $n>3$ it is shown that the distributions appearing form a compact set. An essential role is played by the limit theorem for logarithms of denominators of continued fractions of random numbers.

DOI: https://doi.org/10.4213/rm7150

Full text: PDF file (532 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2007, 62:4, 713–725

Bibliographic databases:

UDC: 517.987.5
MSC: Primary 11N25; Secondary 11A55, 11D04, 11K50, 28DXX, 37AXX
Received: 29.07.2007

Citation: J. Bourgain, Ya. G. Sinai, “Limit behaviour of large Frobenius numbers”, Uspekhi Mat. Nauk, 62:4(376) (2007), 77–90; Russian Math. Surveys, 62:4 (2007), 713–725

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Sinai Ya.G., Ulcigrai C., “Renewal-type limit theorem for the Gauss map and continued fractions”, Ergodic Theory Dynam. Systems, 28:2 (2008), 643–655  crossref  mathscinet  zmath  isi  scopus
    2. A. V. Ustinov, “The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments”, Sb. Math., 200:4 (2009), 597–627  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Ya. G. Sinai, “Nonstandard ergodic theorems for unbounded functions”, Problems Inform. Transmission, 45:4 (2009), 406–409  mathnet  crossref  isi
    4. Aliev I., Henk M., “Integer knapsacks: average behavior of the Frobenius numbers”, Math. Oper. Res., 34:3 (2009), 698–705  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. V. Ustinov, “On the statistical properties of elements of continued fractions”, Dokl. Math., 79:1 (2009), 87–89  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. Shchur V., Sinai Ya., Ustinov A., “Limiting distribution of Frobenius numbers for $n=3$”, J. Number Theory, 129:11 (2009), 2778–2789  crossref  mathscinet  zmath  isi  elib  scopus
    7. Marklof J., “The asymptotic distribution of Frobenius numbers”, Invent. Math., 181:1 (2010), 179–207  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. A. V. Ustinov, “On the distribution of Frobenius numbers with three arguments”, Izv. Math., 74:5 (2010), 1023–1049  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Aliev I., Henk M., Hinrichs A., “Expected Frobenius numbers”, J. Comb. Theory, Ser. A, 118:2 (2011), 525–531  crossref  mathscinet  zmath  isi  scopus
    10. A. V. Ustinov, “Geometric proof of Rødseth's formula for Frobenius numbers”, Proc. Steklov Inst. Math., 276 (2012), 275–282  mathnet  crossref  mathscinet  isi
    11. Shparlinski I.E., “Modular hyperbolas”, Jap. J. Math., 7:2 (2012), 235–294  crossref  mathscinet  zmath  isi  elib  scopus
    12. Strömbergsson A., “On the limit distribution of Frobenius numbers”, Acta Arith., 152:1 (2012), 81–107  crossref  mathscinet  zmath  isi  elib  scopus
    13. Aliev I., Fukshansky L., Henk M., “Generalized Frobenius Numbers: Bounds and Average Behavior”, Acta Arith., 155:1 (2012), 53–62  crossref  mathscinet  zmath  isi  elib  scopus
    14. A. I. Bufetov, B. M. Gurevich, K. M. Khanin, F. Cellarosi, “The Abel Prize award to Ya. G. Sinai”, Russian Math. Surveys, 69:5 (2014), 931–956  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. Han Li, “Effective limit distribution of the Frobenius numbers”, Compositio Math, 2014, 1  crossref  mathscinet  isi  scopus
    16. W.M.. Schmidt, “Integer matrices, sublattices of
      $$\mathbb {Z}^{m}$$
      Z m , and Frobenius numbers”, Monatsh Math, 2015  crossref  mathscinet  isi  scopus
    17. A. V. Ustinov, “Three-dimensional continued fractions and Kloosterman sums”, Russian Math. Surveys, 70:3 (2015), 483–556  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. I. S. Vorob'ev, “On the Frobenius problem for three arguments”, Sb. Math., 207:6 (2016), 816–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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