Three-gap theorem (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Three-gap theorem" in English language version.

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Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", The American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, JSTOR 10.4169/amer.math.monthly.124.8.741, MR 3706822, S2CID 119670663
van Ravenstein, Tony (1988), "The three-gap theorem (Steinhaus conjecture)", Journal of the Australian Mathematical Society, Series A, 45 (3): 360–370, doi:10.1017/S1446788700031062, MR 0957201
Alessandri, Pascal; Berthé, Valérie (1998), "Three distance theorems and combinatorics on words", L'Enseignement mathématique, 44 (1–2): 103–132, MR 1643286; see in particular Section 2.1, "Complexity and frequencies of codings of rotations"
Haynes, Alan; Marklof, Jens (2020), "Higher dimensional Steinhaus and Slater problems via homogeneous dynamics", Annales Scientifiques de l'École Normale Supérieure, 53 (2): 537–557, arXiv:1707.04094, doi:10.24033/asens.2427, MR 4094564, S2CID 67851217, The first proofs of this remarkable fact were published in 1957 by Sós, in 1958 by Surányi, and in 1959 by Świerczkowski
Świerczkowski, S. (1959), "On successive settings of an arc on the circumference of a circle", Fundamenta Mathematicae, 46 (2): 187–189, doi:10.4064/fm-46-2-187-189, MR 0104651
Halton, John H. (1965), "The distribution of the sequence $\{n\xi \}\,(n=0,\,1,\,2,\,\ldots )$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 61 (3): 665–670, doi:10.1017/S0305004100039013, MR 0202668, S2CID 123400321
Chevallier, Nicolas (2007), "Cyclic groups and the three distance theorem", Canadian Journal of Mathematics, 59 (3): 503–552, doi:10.4153/CJM-2007-022-3, MR 2319157, S2CID 123011205
Vijay, Sujith (2008), "Eleven Euclidean distances are enough", Journal of Number Theory, 128 (6): 1655–1661, arXiv:math/0609536, doi:10.1016/j.jnt.2007.08.016, MR 2419185, S2CID 119655772
Bleher, Pavel M.; Homma, Youkow; Ji, Lyndon L.; Roeder, Roland K. W.; Shen, Jeffrey D. (2012), "Nearest neighbor distances on a circle: multidimensional case", Journal of Statistical Physics, 146 (2): 446–465, arXiv:1107.4134, Bibcode:2012JSP...146..446B, doi:10.1007/s10955-011-0367-8, MR 2873022, S2CID 99723
Slater, Noel B. (1967), "Gaps and steps for the sequence $n\theta {\bmod {1}}$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 63 (4): 1115–1123, doi:10.1017/S0305004100042195, MR 0217019, S2CID 121496726
Biringer, Ian; Schmidt, Benjamin (2008), "The three gap theorem and Riemannian geometry", Geometriae Dedicata, 136: 175–190, arXiv:0803.1250, doi:10.1007/s10711-008-9283-8, MR 2443351, S2CID 6389675
Haynes, Alan; Koivusalo, Henna; Walton, James; Sadun, Lorenzo (2016), "Gaps problems and frequencies of patches in cut and project sets" (PDF), Mathematical Proceedings of the Cambridge Philosophical Society, 161 (1): 65–85, Bibcode:2016MPCPS.161...65H, doi:10.1017/S0305004116000128, MR 3505670, S2CID 55686324
Liang, Frank M. (1979), "A short proof of the $3d$ distance theorem", Discrete Mathematics, 28 (3): 325–326, doi:10.1016/0012-365X(79)90140-7, MR 0548632
Shiu, Peter (2018), "A footnote to the three gaps theorem", The American Mathematical Monthly, 125 (3): 264–266, doi:10.1080/00029890.2018.1412210, MR 3768035, S2CID 125810745
Slater, N. B. (1950), "The distribution of the integers $N$ for which $\theta N<\phi$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 46 (4): 525–534, doi:10.1017/S0305004100026086, MR 0041891, S2CID 120454265

arxiv.org

Mayero, Micaela (2000), "The three gap theorem (Steinhaus conjecture)", Types for Proofs and Programs: International Workshop, TYPES'99, Lökeberg, Sweden, June 12–16, 1999, Selected Papers, Lecture Notes in Computer Science, vol. 1956, Springer, pp. 162–173, arXiv:cs/0609124, doi:10.1007/3-540-44557-9_10, ISBN 978-3-540-41517-6, S2CID 3228597
Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", The American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, JSTOR 10.4169/amer.math.monthly.124.8.741, MR 3706822, S2CID 119670663
Akiyama, Shigeki (March 2020), "Spiral Delone sets and three distance theorem", Nonlinearity, 33 (5): 2533–2540, arXiv:1904.10815, Bibcode:2020Nonli..33.2533A, doi:10.1088/1361-6544/ab74ad, S2CID 129945118
Haynes, Alan; Marklof, Jens (2020), "Higher dimensional Steinhaus and Slater problems via homogeneous dynamics", Annales Scientifiques de l'École Normale Supérieure, 53 (2): 537–557, arXiv:1707.04094, doi:10.24033/asens.2427, MR 4094564, S2CID 67851217, The first proofs of this remarkable fact were published in 1957 by Sós, in 1958 by Surányi, and in 1959 by Świerczkowski
Vijay, Sujith (2008), "Eleven Euclidean distances are enough", Journal of Number Theory, 128 (6): 1655–1661, arXiv:math/0609536, doi:10.1016/j.jnt.2007.08.016, MR 2419185, S2CID 119655772
Bleher, Pavel M.; Homma, Youkow; Ji, Lyndon L.; Roeder, Roland K. W.; Shen, Jeffrey D. (2012), "Nearest neighbor distances on a circle: multidimensional case", Journal of Statistical Physics, 146 (2): 446–465, arXiv:1107.4134, Bibcode:2012JSP...146..446B, doi:10.1007/s10955-011-0367-8, MR 2873022, S2CID 99723
Biringer, Ian; Schmidt, Benjamin (2008), "The three gap theorem and Riemannian geometry", Geometriae Dedicata, 136: 175–190, arXiv:0803.1250, doi:10.1007/s10711-008-9283-8, MR 2443351, S2CID 6389675

books.google.com

Allouche, Jean-Paul; Shallit, Jeffrey (2003), "2.6 The Three-Distance Theorem", Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, pp. 53–55, ISBN 9780521823326
Adam, John A. (2011), A Mathematical Nature Walk, Princeton University Press, pp. 35–41, ISBN 9781400832903
Strohm, Reinhard; Blackburn, Bonnie J., eds. (2001), Music as Concept and Practice in the Late Middle Ages, Volume 3, Part 1, New Oxford history of music, Oxford University Press, p. 252, ISBN 9780198162056
Benson, Donald C. (2003), A Smoother Pebble: Mathematical Explorations, Oxford University Press, p. 51, ISBN 9780198032977
Narushima, Terumi (2017), Microtonality and the Tuning Systems of Erv Wilson: Mapping the Harmonic Spectrum, Routledge Studies in Music Theory, Routledge, pp. 90–91, ISBN 9781317513421

bridgesmathart.org

archive.bridgesmathart.org

Haack, Joel K. (1999), "The mathematics of the just intonation used in the music of Terry Riley", in Sarhangi, Reza (ed.), Bridges: Mathematical Connections in Art, Music, and Science, Southwestern College, Winfield, Kansas: Bridges Conference, pp. 101–110, ISBN 0-9665201-1-4

doi.org

Mayero, Micaela (2000), "The three gap theorem (Steinhaus conjecture)", Types for Proofs and Programs: International Workshop, TYPES'99, Lökeberg, Sweden, June 12–16, 1999, Selected Papers, Lecture Notes in Computer Science, vol. 1956, Springer, pp. 162–173, arXiv:cs/0609124, doi:10.1007/3-540-44557-9_10, ISBN 978-3-540-41517-6, S2CID 3228597
Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", The American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, JSTOR 10.4169/amer.math.monthly.124.8.741, MR 3706822, S2CID 119670663
van Ravenstein, Tony (1988), "The three-gap theorem (Steinhaus conjecture)", Journal of the Australian Mathematical Society, Series A, 45 (3): 360–370, doi:10.1017/S1446788700031062, MR 0957201
van Ravenstein, Tony (1987), "Number sequences and phyllotaxis", Bulletin of the Australian Mathematical Society, 36 (2): 333, doi:10.1017/s0004972700026605
Akiyama, Shigeki (March 2020), "Spiral Delone sets and three distance theorem", Nonlinearity, 33 (5): 2533–2540, arXiv:1904.10815, Bibcode:2020Nonli..33.2533A, doi:10.1088/1361-6544/ab74ad, S2CID 129945118
Baroin, Gilles; Calvet, André (2019), "Visualizing temperaments: squaring the circle?", in Montiel, Mariana; Gomez-Martin, Francisco; Agustín-Aquino, Octavio A. (eds.), Mathematics and Computation in Music: 7th International Conference, MCM 2019, Madrid, Spain, June 18–21, 2019, Proceedings, Springer International Publishing, pp. 333–337, doi:10.1007/978-3-030-21392-3_27, S2CID 184482714
Carey, Norman; Clampitt, David (October 1989), "Aspects of well-formed scales", Music Theory Spectrum, 11 (2): 187–206, doi:10.2307/745935, JSTOR 745935
Carey, Norman (2007), "Coherence and sameness in well-formed and pairwise well-formed scales", Journal of Mathematics and Music, 1 (2): 79–98, doi:10.1080/17459730701376743, S2CID 120586231
Haynes, Alan; Marklof, Jens (2020), "Higher dimensional Steinhaus and Slater problems via homogeneous dynamics", Annales Scientifiques de l'École Normale Supérieure, 53 (2): 537–557, arXiv:1707.04094, doi:10.24033/asens.2427, MR 4094564, S2CID 67851217, The first proofs of this remarkable fact were published in 1957 by Sós, in 1958 by Surányi, and in 1959 by Świerczkowski
Świerczkowski, S. (1959), "On successive settings of an arc on the circumference of a circle", Fundamenta Mathematicae, 46 (2): 187–189, doi:10.4064/fm-46-2-187-189, MR 0104651
Halton, John H. (1965), "The distribution of the sequence $\{n\xi \}\,(n=0,\,1,\,2,\,\ldots )$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 61 (3): 665–670, doi:10.1017/S0305004100039013, MR 0202668, S2CID 123400321
Chevallier, Nicolas (2007), "Cyclic groups and the three distance theorem", Canadian Journal of Mathematics, 59 (3): 503–552, doi:10.4153/CJM-2007-022-3, MR 2319157, S2CID 123011205
Vijay, Sujith (2008), "Eleven Euclidean distances are enough", Journal of Number Theory, 128 (6): 1655–1661, arXiv:math/0609536, doi:10.1016/j.jnt.2007.08.016, MR 2419185, S2CID 119655772
Bleher, Pavel M.; Homma, Youkow; Ji, Lyndon L.; Roeder, Roland K. W.; Shen, Jeffrey D. (2012), "Nearest neighbor distances on a circle: multidimensional case", Journal of Statistical Physics, 146 (2): 446–465, arXiv:1107.4134, Bibcode:2012JSP...146..446B, doi:10.1007/s10955-011-0367-8, MR 2873022, S2CID 99723
Slater, Noel B. (1967), "Gaps and steps for the sequence $n\theta {\bmod {1}}$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 63 (4): 1115–1123, doi:10.1017/S0305004100042195, MR 0217019, S2CID 121496726
Biringer, Ian; Schmidt, Benjamin (2008), "The three gap theorem and Riemannian geometry", Geometriae Dedicata, 136: 175–190, arXiv:0803.1250, doi:10.1007/s10711-008-9283-8, MR 2443351, S2CID 6389675
Haynes, Alan; Koivusalo, Henna; Walton, James; Sadun, Lorenzo (2016), "Gaps problems and frequencies of patches in cut and project sets" (PDF), Mathematical Proceedings of the Cambridge Philosophical Society, 161 (1): 65–85, Bibcode:2016MPCPS.161...65H, doi:10.1017/S0305004116000128, MR 3505670, S2CID 55686324
Liang, Frank M. (1979), "A short proof of the $3d$ distance theorem", Discrete Mathematics, 28 (3): 325–326, doi:10.1016/0012-365X(79)90140-7, MR 0548632
Shiu, Peter (2018), "A footnote to the three gaps theorem", The American Mathematical Monthly, 125 (3): 264–266, doi:10.1080/00029890.2018.1412210, MR 3768035, S2CID 125810745
Slater, N. B. (1950), "The distribution of the integers $N$ for which $\theta N<\phi$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 46 (4): 525–534, doi:10.1017/S0305004100026086, MR 0041891, S2CID 120454265

dur.ac.uk

dro.dur.ac.uk

Haynes, Alan; Koivusalo, Henna; Walton, James; Sadun, Lorenzo (2016), "Gaps problems and frequencies of patches in cut and project sets" (PDF), Mathematical Proceedings of the Cambridge Philosophical Society, 161 (1): 65–85, Bibcode:2016MPCPS.161...65H, doi:10.1017/S0305004116000128, MR 3505670, S2CID 55686324

handle.net

hdl.handle.net

Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", The American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, JSTOR 10.4169/amer.math.monthly.124.8.741, MR 3706822, S2CID 119670663

harvard.edu

ui.adsabs.harvard.edu

Akiyama, Shigeki (March 2020), "Spiral Delone sets and three distance theorem", Nonlinearity, 33 (5): 2533–2540, arXiv:1904.10815, Bibcode:2020Nonli..33.2533A, doi:10.1088/1361-6544/ab74ad, S2CID 129945118
Bleher, Pavel M.; Homma, Youkow; Ji, Lyndon L.; Roeder, Roland K. W.; Shen, Jeffrey D. (2012), "Nearest neighbor distances on a circle: multidimensional case", Journal of Statistical Physics, 146 (2): 446–465, arXiv:1107.4134, Bibcode:2012JSP...146..446B, doi:10.1007/s10955-011-0367-8, MR 2873022, S2CID 99723
Haynes, Alan; Koivusalo, Henna; Walton, James; Sadun, Lorenzo (2016), "Gaps problems and frequencies of patches in cut and project sets" (PDF), Mathematical Proceedings of the Cambridge Philosophical Society, 161 (1): 65–85, Bibcode:2016MPCPS.161...65H, doi:10.1017/S0305004116000128, MR 3505670, S2CID 55686324

irif.fr

Alessandri, Pascal; Berthé, Valérie (1998), "Three distance theorems and combinatorics on words", L'Enseignement mathématique, 44 (1–2): 103–132, MR 1643286; see in particular Section 2.1, "Complexity and frequencies of codings of rotations"

jstor.org

Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", The American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, JSTOR 10.4169/amer.math.monthly.124.8.741, MR 3706822, S2CID 119670663
Carey, Norman; Clampitt, David (October 1989), "Aspects of well-formed scales", Music Theory Spectrum, 11 (2): 187–206, doi:10.2307/745935, JSTOR 745935

semanticscholar.org

api.semanticscholar.org

Mayero, Micaela (2000), "The three gap theorem (Steinhaus conjecture)", Types for Proofs and Programs: International Workshop, TYPES'99, Lökeberg, Sweden, June 12–16, 1999, Selected Papers, Lecture Notes in Computer Science, vol. 1956, Springer, pp. 162–173, arXiv:cs/0609124, doi:10.1007/3-540-44557-9_10, ISBN 978-3-540-41517-6, S2CID 3228597
Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", The American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, JSTOR 10.4169/amer.math.monthly.124.8.741, MR 3706822, S2CID 119670663
Akiyama, Shigeki (March 2020), "Spiral Delone sets and three distance theorem", Nonlinearity, 33 (5): 2533–2540, arXiv:1904.10815, Bibcode:2020Nonli..33.2533A, doi:10.1088/1361-6544/ab74ad, S2CID 129945118
Baroin, Gilles; Calvet, André (2019), "Visualizing temperaments: squaring the circle?", in Montiel, Mariana; Gomez-Martin, Francisco; Agustín-Aquino, Octavio A. (eds.), Mathematics and Computation in Music: 7th International Conference, MCM 2019, Madrid, Spain, June 18–21, 2019, Proceedings, Springer International Publishing, pp. 333–337, doi:10.1007/978-3-030-21392-3_27, S2CID 184482714
Carey, Norman (2007), "Coherence and sameness in well-formed and pairwise well-formed scales", Journal of Mathematics and Music, 1 (2): 79–98, doi:10.1080/17459730701376743, S2CID 120586231
Haynes, Alan; Marklof, Jens (2020), "Higher dimensional Steinhaus and Slater problems via homogeneous dynamics", Annales Scientifiques de l'École Normale Supérieure, 53 (2): 537–557, arXiv:1707.04094, doi:10.24033/asens.2427, MR 4094564, S2CID 67851217, The first proofs of this remarkable fact were published in 1957 by Sós, in 1958 by Surányi, and in 1959 by Świerczkowski
Halton, John H. (1965), "The distribution of the sequence $\{n\xi \}\,(n=0,\,1,\,2,\,\ldots )$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 61 (3): 665–670, doi:10.1017/S0305004100039013, MR 0202668, S2CID 123400321
Chevallier, Nicolas (2007), "Cyclic groups and the three distance theorem", Canadian Journal of Mathematics, 59 (3): 503–552, doi:10.4153/CJM-2007-022-3, MR 2319157, S2CID 123011205
Vijay, Sujith (2008), "Eleven Euclidean distances are enough", Journal of Number Theory, 128 (6): 1655–1661, arXiv:math/0609536, doi:10.1016/j.jnt.2007.08.016, MR 2419185, S2CID 119655772
Bleher, Pavel M.; Homma, Youkow; Ji, Lyndon L.; Roeder, Roland K. W.; Shen, Jeffrey D. (2012), "Nearest neighbor distances on a circle: multidimensional case", Journal of Statistical Physics, 146 (2): 446–465, arXiv:1107.4134, Bibcode:2012JSP...146..446B, doi:10.1007/s10955-011-0367-8, MR 2873022, S2CID 99723
Slater, Noel B. (1967), "Gaps and steps for the sequence $n\theta {\bmod {1}}$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 63 (4): 1115–1123, doi:10.1017/S0305004100042195, MR 0217019, S2CID 121496726
Biringer, Ian; Schmidt, Benjamin (2008), "The three gap theorem and Riemannian geometry", Geometriae Dedicata, 136: 175–190, arXiv:0803.1250, doi:10.1007/s10711-008-9283-8, MR 2443351, S2CID 6389675
Haynes, Alan; Koivusalo, Henna; Walton, James; Sadun, Lorenzo (2016), "Gaps problems and frequencies of patches in cut and project sets" (PDF), Mathematical Proceedings of the Cambridge Philosophical Society, 161 (1): 65–85, Bibcode:2016MPCPS.161...65H, doi:10.1017/S0305004116000128, MR 3505670, S2CID 55686324
Shiu, Peter (2018), "A footnote to the three gaps theorem", The American Mathematical Monthly, 125 (3): 264–266, doi:10.1080/00029890.2018.1412210, MR 3768035, S2CID 125810745
Slater, N. B. (1950), "The distribution of the integers $N$ for which $\theta N<\phi$ ", Mathematical Proceedings of the Cambridge Philosophical Society, 46 (4): 525–534, doi:10.1017/S0305004100026086, MR 0041891, S2CID 120454265

univ-mlv.fr

www-igm.univ-mlv.fr

Lothaire, M. (2002), "Sturmian Words", Algebraic Combinatorics on Words, Cambridge: Cambridge University Press, pp. 40–97, ISBN 978-0-521-81220-7, Zbl 1001.68093. Lothaire uses the property of having $d+1$ words of length $d$ as a definition of Sturmian words, rather than as a consequence of the definition. For the equivalence of this property with the definition stated here, see Theorem 2.1.13, p. 51. For the three frequencies of these words see Theorem 2.2.37, p. 73.

zbmath.org

Lothaire, M. (2002), "Sturmian Words", Algebraic Combinatorics on Words, Cambridge: Cambridge University Press, pp. 40–97, ISBN 978-0-521-81220-7, Zbl 1001.68093. Lothaire uses the property of having $d+1$ words of length $d$ as a definition of Sturmian words, rather than as a consequence of the definition. For the equivalence of this property with the definition stated here, see Theorem 2.1.13, p. 51. For the three frequencies of these words see Theorem 2.2.37, p. 73.