Transcendental number (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Transcendental number" in English language version.

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Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly, 107 (5): 448–449, doi:10.2307/2695302, JSTOR 2695302, MR 1763399
Blanchard & Mendès France 1982 Blanchard, André; Mendès France, Michel (1982). "Symétrie et transcendance". Bulletin des Sciences Mathématiques. 106 (3): 325–335. MR 0680277.

archive.org

Leibniz, Gerhardt & Pertz 1858, pp. 97–98; Bourbaki 1994, p. 74 Leibniz, G.W.; Gerhardt, Karl Immanuel; Pertz, Georg Heinrich (1858). Leibnizens mathematische Schriften. Vol. 5. A. Asher & Co. pp. 97–98 – via Internet Archive. Bourbaki, N. (1994). Elements of the History of Mathematics. Springer. ISBN 9783540647676 – via Internet Archive.
Steven R. Finch (2003). Mathematical Constants. Cambridge University Press. p. 479. ISBN 978-3-540-67695-9. Schmutz.

arxiv.org

Heuer, Nicolaus; Loeh, Clara (1 November 2019). "Transcendental simplicial volumes". arXiv:1911.06386 [math.GT].
Adamczewski & Bugeaud 2005 Adamczewski, Boris; Bugeaud, Yann (2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica. 195 (1): 1–20. arXiv:math/0511677. Bibcode:2005math.....11677A. doi:10.1007/BF02588048. S2CID 15521751.
Mező, István; Baricz, Árpád (June 22, 2015). "On the generalization of the Lambert W function". arXiv:1408.3999 [math.CA].
Yoshinaga, Masahiko (2008-05-03). "Periods and elementary real numbers". arXiv:0805.0349 [math.AG].
Adamczewski, Boris (March 2013). "The Many Faces of the Kempner Number". arXiv:1303.1685 [math.NT].
Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X. ISSN 0273-0979.

books.google.com

Bunday, B. D.; Mulholland, H. (20 May 2014). Pure Mathematics for Advanced Level. Butterworth-Heinemann. ISBN 978-1-4831-0613-7. Retrieved 21 March 2021.

britannica.com

"Real number". Encyclopædia Britannica. mathematics. Retrieved 2020-08-11.

cambridge.org

"Mathematical constants". Mathematics (general). Cambridge University Press. Retrieved 2022-09-22.

digizeitschriften.de

Cantor 1874; Gray 1994 Cantor, G. (1874). "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen". J. Reine Angew. Math. 77: 258–262. Gray, Robert (1994). "Georg Cantor and transcendental numbers". Amer. Math. Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Zbl 0827.01004 – via maa.org.
Cantor 1878, p. 254 Cantor, G. (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". J. Reine Angew. Math. 84: 242–258.

diku.dk

hjemmesider.diku.dk

Grue Simonsen, Jakob. "Specker Sequences Revisited" (PDF). hjemmesider.diku.dk.

doi.org

Baker, A. (1964). "On Mahler's classification of transcendental numbers". Acta Mathematica. 111: 97–120. doi:10.1007/bf02391010. S2CID 122023355.
Erdős & Dudley 1983 Erdős, P.; Dudley, U. (1983). "Some Remarks and Problems in Number Theory Related to the Work of Euler" (PDF). Mathematics Magazine. 56 (5): 292–298. CiteSeerX 10.1.1.210.6272. doi:10.2307/2690369. JSTOR 2690369.
Kempner 1916 Kempner, Aubrey J. (1916). "On Transcendental Numbers". Transactions of the American Mathematical Society. 17 (4): 476–482. doi:10.2307/1988833. JSTOR 1988833.
Cantor 1874; Gray 1994 Cantor, G. (1874). "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen". J. Reine Angew. Math. 77: 258–262. Gray, Robert (1994). "Georg Cantor and transcendental numbers". Amer. Math. Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Zbl 0827.01004 – via maa.org.
Adamczewski & Bugeaud 2005 Adamczewski, Boris; Bugeaud, Yann (2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica. 195 (1): 1–20. arXiv:math/0511677. Bibcode:2005math.....11677A. doi:10.1007/BF02588048. S2CID 15521751.
Siegel, Carl L. (2014). "Über einige Anwendungen diophantischer Approximationen: Abhandlungen der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1929, Nr. 1". On Some Applications of Diophantine Approximations (in German). Scuola Normale Superiore. pp. 81–138. doi:10.1007/978-88-7642-520-2_2. ISBN 978-88-7642-520-2.
Lorch, Lee; Muldoon, Martin E. (1995). "Transcendentality of zeros of higher dereivatives of functions involving Bessel functions". International Journal of Mathematics and Mathematical Sciences. 18 (3): 551–560. doi:10.1155/S0161171295000706.
Davison & Shallit 1991 Davison, J. Les; Shallit, J.O. (1991). "Continued fractions for some alternating series". Monatshefte für Mathematik. 111 (2): 119–126. doi:10.1007/BF01332350. S2CID 120003890.
Duverney, Daniel; Nishioka, Keiji; Nishioka, Kumiko; Shiokawa, Iekata (1997). "Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 73 (7): 140–142. doi:10.3792/pjaa.73.140. ISSN 0386-2194.
Bertrand, Daniel (1997). "Theta functions and transcendence". The Ramanujan Journal. 1 (4): 339–350. doi:10.1023/A:1009749608672. S2CID 118628723.
Todd, John (1975). "The lemniscate constants". Communications of the ACM. 18: 14–19. doi:10.1145/360569.360580. S2CID 85873.
Kurosawa, Takeshi (2007-03-01). "Transcendence of certain series involving binary linear recurrences". Journal of Number Theory. 123 (1): 35–58. doi:10.1016/j.jnt.2006.05.019. ISSN 0022-314X.
Mahler 1929; Allouche & Shallit 2003, p. 387 Mahler, K. (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929. Allouche, J.-P. [in French]; Shallit, J. (2003). Automatic Sequences: Theory, applications, generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly, 107 (5): 448–449, doi:10.2307/2695302, JSTOR 2695302, MR 1763399
Mahler, Kurt; Mordell, Louis Joel (1968-06-04). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.
Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X. ISSN 0273-0979.
Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata (2012-09-01). "Algebraic independence of certain numbers related to modular functions". Functiones et Approximatio Commentarii Mathematici. 47 (1). doi:10.7169/facm/2012.47.1.10. ISSN 0208-6573.
Brownawell, W. Dale (1974-02-01). "The algebraic independence of certain numbers related by the exponential function". Journal of Number Theory. 6: 22–31. doi:10.1016/0022-314X(74)90005-5. ISSN 0022-314X.
Murty, M. Ram; Saradha, N. (2010-12-01). "Euler–Lehmer constants and a conjecture of Erdös". Journal of Number Theory. 130 (12): 2671–2682. doi:10.1016/j.jnt.2010.07.004. ISSN 0022-314X.
Murty, M. Ram; Zaytseva, Anastasia (2013-01-01). "Transcendence of generalized Euler constants". The American Mathematical Monthly. 120 (1): 48–54. doi:10.4169/amer.math.monthly.120.01.048. ISSN 0002-9890. S2CID 20495981.
Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.
Rivoal, T.; Zudilin, W. (2003-08-01). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen. 326 (4): 705–721. doi:10.1007/s00208-003-0420-2. hdl:1959.13/803688. ISSN 1432-1807. S2CID 59328860.

handle.net

hdl.handle.net

Rivoal, T.; Zudilin, W. (2003-08-01). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen. 326 (4): 705–721. doi:10.1007/s00208-003-0420-2. hdl:1959.13/803688. ISSN 1432-1807. S2CID 59328860.

harvard.edu

ui.adsabs.harvard.edu

Adamczewski & Bugeaud 2005 Adamczewski, Boris; Bugeaud, Yann (2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica. 195 (1): 1–20. arXiv:math/0511677. Bibcode:2005math.....11677A. doi:10.1007/BF02588048. S2CID 15521751.
Mahler, Kurt; Mordell, Louis Joel (1968-06-04). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.
Brownawell, W. Dale (1974-02-01). "The algebraic independence of certain numbers related by the exponential function". Journal of Number Theory. 6: 22–31. doi:10.1016/0022-314X(74)90005-5. ISSN 0022-314X.

imj-prg.fr

webusers.imj-prg.fr

Waldschmidt, Michel (September 7, 2005). "Transcendence of Periods: The State of the Art" (PDF). webusers.imj-prg.fr.

jstor.org

Erdős & Dudley 1983 Erdős, P.; Dudley, U. (1983). "Some Remarks and Problems in Number Theory Related to the Work of Euler" (PDF). Mathematics Magazine. 56 (5): 292–298. CiteSeerX 10.1.1.210.6272. doi:10.2307/2690369. JSTOR 2690369.
Kempner 1916 Kempner, Aubrey J. (1916). "On Transcendental Numbers". Transactions of the American Mathematical Society. 17 (4): 476–482. doi:10.2307/1988833. JSTOR 1988833.
Cantor 1874; Gray 1994 Cantor, G. (1874). "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen". J. Reine Angew. Math. 77: 258–262. Gray, Robert (1994). "Georg Cantor and transcendental numbers". Amer. Math. Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Zbl 0827.01004 – via maa.org.
Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly, 107 (5): 448–449, doi:10.2307/2695302, JSTOR 2695302, MR 1763399

maa.org

Cantor 1874; Gray 1994 Cantor, G. (1874). "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen". J. Reine Angew. Math. 77: 258–262. Gray, Robert (1994). "Georg Cantor and transcendental numbers". Amer. Math. Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Zbl 0827.01004 – via maa.org.

mathdoc.fr

sites.mathdoc.fr

Liouville 1851 Liouville, J. (1851). "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques" (PDF). J. Math. Pures Appl. 16: 133–142.

oed.com

"transcendental". Oxford English Dictionary. s.v.

oeis.org

"A140654 - OEIS". oeis.org. Retrieved 2023-08-12.
"A143347 - OEIS". oeis.org. Retrieved 2023-08-09.
"A093540 - OEIS". oeis.org. Retrieved 2023-08-12.

projecteuclid.org

Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.

psu.edu

citeseerx.ist.psu.edu

Erdős & Dudley 1983 Erdős, P.; Dudley, U. (1983). "Some Remarks and Problems in Number Theory Related to the Work of Euler" (PDF). Mathematics Magazine. 56 (5): 292–298. CiteSeerX 10.1.1.210.6272. doi:10.2307/2690369. JSTOR 2690369.

renyi.hu

users.renyi.hu

Erdős & Dudley 1983 Erdős, P.; Dudley, U. (1983). "Some Remarks and Problems in Number Theory Related to the Work of Euler" (PDF). Mathematics Magazine. 56 (5): 292–298. CiteSeerX 10.1.1.210.6272. doi:10.2307/2690369. JSTOR 2690369.

royalsocietypublishing.org

Mahler, Kurt; Mordell, Louis Joel (1968-06-04). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.

semanticscholar.org

api.semanticscholar.org

Baker, A. (1964). "On Mahler's classification of transcendental numbers". Acta Mathematica. 111: 97–120. doi:10.1007/bf02391010. S2CID 122023355.
Adamczewski & Bugeaud 2005 Adamczewski, Boris; Bugeaud, Yann (2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica. 195 (1): 1–20. arXiv:math/0511677. Bibcode:2005math.....11677A. doi:10.1007/BF02588048. S2CID 15521751.
Davison & Shallit 1991 Davison, J. Les; Shallit, J.O. (1991). "Continued fractions for some alternating series". Monatshefte für Mathematik. 111 (2): 119–126. doi:10.1007/BF01332350. S2CID 120003890.
Bertrand, Daniel (1997). "Theta functions and transcendence". The Ramanujan Journal. 1 (4): 339–350. doi:10.1023/A:1009749608672. S2CID 118628723.
Todd, John (1975). "The lemniscate constants". Communications of the ACM. 18: 14–19. doi:10.1145/360569.360580. S2CID 85873.
Mahler 1929; Allouche & Shallit 2003, p. 387 Mahler, K. (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929. Allouche, J.-P. [in French]; Shallit, J. (2003). Automatic Sequences: Theory, applications, generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
Mahler, Kurt; Mordell, Louis Joel (1968-06-04). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.
Murty, M. Ram; Zaytseva, Anastasia (2013-01-01). "Transcendence of generalized Euler constants". The American Mathematical Monthly. 120 (1): 48–54. doi:10.4169/amer.math.monthly.120.01.048. ISSN 0002-9890. S2CID 20495981.
Rivoal, T.; Zudilin, W. (2003-08-01). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen. 326 (4): 705–721. doi:10.1007/s00208-003-0420-2. hdl:1959.13/803688. ISSN 1432-1807. S2CID 59328860.

springer.com

link.springer.com

Bertrand, Daniel (1997). "Theta functions and transcendence". The Ramanujan Journal. 1 (4): 339–350. doi:10.1023/A:1009749608672. S2CID 118628723.

st-andrews.ac.uk

www-history.mcs.st-andrews.ac.uk

Baker, Alan (1998). J.J. O'Connor and E.F. Robertson. www-history.mcs.st-andrews.ac.uk (biographies). The MacTutor History of Mathematics archive. St. Andrew's, Scotland: University of St. Andrew's.

wikipedia.org

fr.wikipedia.org

Allouche & Shallit 2003, pp. 385, 403 Allouche, J.-P. [in French]; Shallit, J. (2003). Automatic Sequences: Theory, applications, generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
Mahler 1929; Allouche & Shallit 2003, p. 387 Mahler, K. (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929. Allouche, J.-P. [in French]; Shallit, J. (2003). Automatic Sequences: Theory, applications, generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.

wisc.edu

sprott.physics.wisc.edu

Pickover, Cliff. "The 15 most famous transcendental numbers". sprott.physics.wisc.edu. Retrieved 2020-01-23.

wolfram.com

mathworld.wolfram.com

"Weisstein, Eric W. "Liouville's Constant", MathWorld".
Weisstein, Eric W. "Dottie Number". Wolfram MathWorld. Wolfram Research, Inc. Retrieved 23 July 2016.
le Lionnais 1979, p. 46 via Wolfram Mathworld, Transcendental Number le Lionnais, F. (1979). Les nombres remarquables. Hermann. ISBN 2-7056-1407-9.
Chudnovsky 1984 via Wolfram Mathworld, Transcendental Number Chudnovsky, G.V. (1984). Contributions to the Theory of Transcendental Numbers. American Mathematical Society. ISBN 978-0-8218-1500-7.
Weisstein, Eric W. "Transcendental Number". mathworld.wolfram.com. Retrieved 2023-08-09.
Weisstein, Eric W. "van der Corput's Constant". mathworld.wolfram.com. Retrieved 2023-08-10.
Weisstein, Eric W. "Zolotarev-Schur Constant". mathworld.wolfram.com. Retrieved 2023-08-12.
Weisstein, Eric W. "Rabbit Constant". mathworld.wolfram.com. Retrieved 2023-08-09.
Weisstein, Eric W. "Weierstrass Constant". mathworld.wolfram.com. Retrieved 2023-08-12.
Weisstein, Eric W. "Irrational Number". MathWorld.
Weisstein, Eric W. "e". mathworld.wolfram.com. Retrieved 2023-08-12.

worldcat.org

Duverney, Daniel; Nishioka, Keiji; Nishioka, Kumiko; Shiokawa, Iekata (1997). "Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 73 (7): 140–142. doi:10.3792/pjaa.73.140. ISSN 0386-2194.
Kurosawa, Takeshi (2007-03-01). "Transcendence of certain series involving binary linear recurrences". Journal of Number Theory. 123 (1): 35–58. doi:10.1016/j.jnt.2006.05.019. ISSN 0022-314X.
Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X. ISSN 0273-0979.
Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata (2012-09-01). "Algebraic independence of certain numbers related to modular functions". Functiones et Approximatio Commentarii Mathematici. 47 (1). doi:10.7169/facm/2012.47.1.10. ISSN 0208-6573.
Brownawell, W. Dale (1974-02-01). "The algebraic independence of certain numbers related by the exponential function". Journal of Number Theory. 6: 22–31. doi:10.1016/0022-314X(74)90005-5. ISSN 0022-314X.
Murty, M. Ram; Saradha, N. (2010-12-01). "Euler–Lehmer constants and a conjecture of Erdös". Journal of Number Theory. 130 (12): 2671–2682. doi:10.1016/j.jnt.2010.07.004. ISSN 0022-314X.
Murty, M. Ram; Zaytseva, Anastasia (2013-01-01). "Transcendence of generalized Euler constants". The American Mathematical Monthly. 120 (1): 48–54. doi:10.4169/amer.math.monthly.120.01.048. ISSN 0002-9890. S2CID 20495981.
Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.
Rivoal, T.; Zudilin, W. (2003-08-01). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen. 326 (4): 705–721. doi:10.1007/s00208-003-0420-2. hdl:1959.13/803688. ISSN 1432-1807. S2CID 59328860.

zbmath.org

Cantor 1874; Gray 1994 Cantor, G. (1874). "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen". J. Reine Angew. Math. 77: 258–262. Gray, Robert (1994). "Georg Cantor and transcendental numbers". Amer. Math. Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Zbl 0827.01004 – via maa.org.
Mahler 1937; Mahler 1976, p. 12 Mahler, K. (1937). "Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen". Proc. Konin. Neder. Akad. Wet. Ser. A (40): 421–428. Mahler, K. (1976). Lectures on Transcendental Numbers. Lecture Notes in Mathematics. Vol. 546. Springer. ISBN 978-3-540-07986-6. Zbl 0332.10019.
Calude 2002, p. 239 Calude, Cristian S. (2002). Information and Randomness: An algorithmic perspective. Texts in Theoretical Computer Science (2nd rev. and ext. ed.). Springer. ISBN 978-3-540-43466-5. Zbl 1055.68058.
Allouche & Shallit 2003, pp. 385, 403 Allouche, J.-P. [in French]; Shallit, J. (2003). Automatic Sequences: Theory, applications, generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
Loxton 1988 Loxton, J.H. (1988). "13. Automata and transcendence". In Baker, A. (ed.). New Advances in Transcendence Theory. Cambridge University Press. pp. 215–228. ISBN 978-0-521-33545-4. Zbl 0656.10032.
Mahler 1929; Allouche & Shallit 2003, p. 387 Mahler, K. (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929. Allouche, J.-P. [in French]; Shallit, J. (2003). Automatic Sequences: Theory, applications, generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
Pytheas Fogg 2002 Pytheas Fogg, N. (2002). Berthé, V.; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Springer. ISBN 978-3-540-44141-0. Zbl 1014.11015.
Bugeaud 2012, p. 113. Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge University Press. ISBN 978-0-521-11169-0. Zbl 1260.11001.