Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Forbidden Integer Ratios of Consecutive Power Sums Read chapter Read first chapter

2016 | OriginalPaper | Chapter

On Liouville Numbers: Yet Another Application of Functional Analysis to Number Theory

Author : Jörn Steuding

Published in: From Arithmetic to Zeta-Functions

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In 1962, Erdős proved that every real number can be represented as a sum and as a product of two Liouville numbers. This has been generalized by Rieger and Schwarz. In this note we shall give an analysis of these results and their proofs. Moreover, we consider a certain subclass of Liouville numbers and prove similar results for this subclass. Since Wolfgang Schwarz had been very much interested in the history of mathematics, and the author shares this interest, he could not resist to include a few historical remarks (and footnotes) on transcendental numbers and Baire’s category theorem which might be interesting for the reader.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter The GCD of the Shifted Fibonacci Sequence
next chapter Natural Boundaries of Power Series with Multiplicative Coefficients in Algebraic Number Fields
Footnotes
1
Which he commented with his famous words “omnem rationem transcendunt”.
 
2
This final version appeared in [56]. His proof relies on an application of Rolle’s theorem to the minimal polynomial of ξ.
 
3
Hermite [39, p. 77]; English translation: “In this context, M. Liouville already obtained a remarkable theorem which is object of his work entitled: Sur des classes très-étendus de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationelles algébrique, and I also remind that the illustrious geometer was the first to show the proposition which is subject of this research for the case of an equation of second degree and of an equation of biquadratic degree”.
 
4
Who obtained his habilitation and venia legendi in 1877 at the University of Würzburg.
 
5
cf. Lützen [57, p. 514].
 
6
Liouville [56, p. 140]; English translation: “I think I can remember that one finds a theorem of this kind announced in a letter of Goldbach to Euler; however, I would not know whether its proof was ever given”.
 
7
Cantor [15, p. 259]; English translation: “(…) so I found the significant difference between a so-called continuum and the epitome of all kind of real algebraic numbers.”
 
8
It might be interesting to notice that Cantor’s doctorate entitled De aequationibus secundi gradus indeterminatis from 1867 dealed with quadratic forms and was supervised by Ernst Eduard Kummer.
 
9
Which was one of Wolfgang Schwarz’s favourite readings. Theodor Schneider was the thesis advisor of Wolfgang Schwarz; he is famous for solving the seventh Hilbert problem on the transcendence of α β for algebraic β and α ≠ 0, 1 simultaneously with and independently to Aleksandar Gelfond in 1934.
 
10
Erdős [27, p. 59].
 
11
As indicated by his warm-hearted obituary [28].
 
12
Erdős [27, p. 60].
 
13
A set M is said to be dense in a topological space X if every point of X is either an element of M or a limit point of M; a set is called nowhere dense if its closure has no interior points.
 
14
A very recommendable monograph on this topic is the book by Oxtoby [67] studying both subjects simultaneously.
 
15
Actually, Borel’s original proof of the statement that almost all real numbers are normal to any base (in the sense of Lebesgue measure) was based on a strong law of large numbers and faulty; it has been corrected by Cantelli and is now known as the Borel–Cantelli lemma. The first rigorous proof of Borel’s statement about normal numbers is due to Georg Faber [30] and Felix Hausdorff [38], respectively. We refer to Barone and Novikoff [8] for details about the interesting polemic between Borel and Cantor’s pupil Felix Bernstein.
 
16
See Oxtoby’s textbook [67, Chap. 19], for further details.
 
17
Building on previous work by Hugo Steinhaus [78].
 
18
See Volkert [82] for a historical discussion of Weierstrass’ construction and its generalization.
 
19
cf. Ilgauds and Purkert [45, p. 47], resp. Maor [59, p. 55]; English translation: “(…) so I protest against using an infinite quantity as accomplished which is nowhere allowed in mathematics. The infinite is only a façon de parler.” Here the word ‘façon de parler’ can be translated literally as ‘manner of speaking’.
 
20
Dugac [23, p. 317]; English translation: “By the way, there is Baire who was the first in France to use Georg Cantor’s works in a systematic way in exactly the general topology of infinity. A. Denjoy told us that ‘there is no doubt’ and that ‘Cantor’s influence on René Baire was substantial’. In particular when it is about infinity ‘Cantor is present in Baire’s work’. On the contrary ‘Baire himself never appealed to the hypothesis that the real numbers form a well-ordered set. He always believed them to be countable”.
 
21
Deiser [21, p. 494]; English translation: “The nowhere dense ([middle third] Cantor set had already been considered by Cantor in 1883, moreover, one can read his proof of the uncountability of \(\mathbb{R}\), first discussed with Dedekind in a letter, as a proof of the category theorem. However, it was Baire to emphasize the concept”.
 
22
Which is constructed by removing each middle third of the unit interval and all remaining intervals.
 
23
For this and the rather difficult relation between the latter mathematician and Cantor see Laugwitz [53].
 
24
Cavaillès and Noether [19]; English translation: “Now we can think of a number, I will call it η, which lies in the interior of all these intervals; it follows easily that this number η is not contained in any of our sequences, henceforth our assumptions was false”.
 
25
German Mathematicians Association; Cantor was one of the founders of this society in 1890.
 
26
van Vleck [79, p. 189]; in the original the names of Osgood and Baire are written in small capitals as well.
 
27
In the case of Osgood and van Vleck both studied and worked with Felix Klein in Göttingen.
 
28
See Barone and Novikoff [7] for details.
 
29
van Vleck [81, p. 7].
 
30
Scheeffer [69, p. 291]; English translation: “Let P be an arbitrary perfect set which is not dense in any interval, and R an arbitrary countable set. Subtracting a constant a from all elements of P, the values x − a form a new set P a . Then one can always determine the constant a in between given bounds such that the set P a contains no value from the set R”. The word ‘Hülfssatz’ is old-fashioned German for a proposition.
 
31
It is remarkable that Bagemihl’s paper [2] was published in the same journal as Erdős’ article [27] on Liouville numbers about a decade earlier; it might be even more remarkable that in this second issue of the Michigan Mathematical Journal there is a paper by Gerald Mac Lane [58] on a conjecture by Erdős and others published on the previous pages. However, taking more than 1500 publications of Erdős and their deep impact into account this might have no meaning at all. Scheeffer’s theorem is also mentioned in the at that time rather influential monograph [76] by Sierpiński.
 
32
The author has learned about Scheeffer’s work and all the other details mentioned above by the blog mathoverflow and the valuable comments of Andres Caicedo on the question entitled ‘A translation of the Cantor set contained in the irrationals’, posted February 22, 2014, and based on Wilman Brito’s book [11].
 
33
Cantor [17, p. 199]; engl. translation: “A sound scholarliness connected with plenty of own thoughts, which with exemplary simplicity, clearness, and elegant language are constituted, are the essential character of his productions. Of his beautiful results, to which he was guided by his brassbound diligence driven investigations, I would like to accent here the proof of the following theorem from No. 4:
“If one knows about a continuous function of one variable that its differential quotient is zero for all values of an interval with exception of those which constitute an arbitrarily given infinite set of first cardinality, then the function is a constant in this interval.”” The words in italics are also in the original in italics.
 
34
Scheeffer [69] uses with ‘Inhalt Null’ (meaning ‘content zero’) another notion introduced by Cantor in place of ‘erster Mächtigkeit’.
 
35
Dugac [23, p. 349]; English translation: “His note does not seem to be very original, in particular since his example had been given 1885 by Ludwig Scheeffer in his treatise in Acta Mathematica (Tome V), and to which I refer to in the last chapter of my thesis. Only the presentation differs and I ask myself whether there could have been a simple coincidence. The note had been communicated by M. Picard after the return from his journey; I could offer to talk to him, however, I had completely forgotten at my last visit; I could talk to him about that another time.” In the author’s opinion there are different interpretations of the notion ‘je me demande s’il peut y avoir là une simple coïncidence’. possible here.
 
36
cf. Weber [84, p. 15]; English translation: “God made natural numbers, all else is the work of man”.
 
37
Hilbert [42, p. 170]; English translation: “No one shall expel us from the paradise that Cantor has created”.
 
38
It might be interesting to notice that his article has the subtitle “Aus einem an Herrn Hilbert gerichteten Briefe”, and that he wrote further “Die Idee (…) verdanke ich Herrn Erhard Schmidt”. meaning that the article contains material from a letter written to Hilbert and that the idea behind is due to Schmidt.
 
39
“Hence, Zermelo rather than Cantor should be regarded as the creator of abstract set theory” wrote Kanamori [48, p. 15].
 
40
The appearing brackets point to publications of the three authors indicating the year and page.
 
41
For a change of thinking credit should be given to, among others, Hausdorff and the Polish school of functional analysts.
 
42
cf. Gray [34, p. 258].
 
43
Another much simpler example provides the middle third Cantor set \(\mathcal{C}\). It is a folklore theorem that every real number can be written as a sum of an integer and two elements of \(\mathcal{C}\).
 
44
Schwarz [72, p. 1]; English translation: “The course is thus a medium between a history of number theory and a report on number-theoretical results. One aim of historical investigations to give orientation could therefore be achieved.” The boldfaced words are as in the original.
 
Literature
1.
2.
3.
F. Bagemihl, W. Seidel, A general principle involving Baire category, with applications to function theory and other fields. Proc. Natl. Acad. Sci. USA 39, 1068–1075 (1953)MathSciNetCrossRefMATH
4.
F. Bagemihl, P. Erdős, W. Seidel, Sur quelques propriétés frontières des fonctions holomorphes définies par certains produits dans le cercle-unité. Ann. Sci. Éc. Norm. Super. 70, 135–147 (1953)
5.
R.L. Baire, Sur les fonctions de variables réelles. Ann. di Mat. Pura ed applicata 3, 1–123 (1899)CrossRefMATH
6.
S. Banach, Über die Baire’sche Kategorie gewisser Funktionenmengen. Stud. Math. 3, 174–179 (1931)MATH
7.
J. Barone, A. Novikoff, The Borel law of normal numbers, the Borel zero-one law, and the work of van Vleck. Hist. Math. 4, 43–65 (1977)MathSciNetCrossRefMATH
8.
J. Barone, A. Novikoff, A history of the axiomatic formulation of probability from Borel to Kolmogorov: Part I. Arch. Hist. Exact Sci. 18, 123–190 (1978)MathSciNetMATH
9.
C.E. Blair, The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. 25, 933–934 (1977)MathSciNetMATH
10.
É. Borel, Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Matematico di Palermo 27, 247–271 (1909)CrossRefMATH
11.
W. Brito, El Teorema de Categoría de Baire y aplicaciones (Universidad de los Andes, Venezuela, 2011)
12.
13.
14.
E.B. Burger, R. Tubbs, Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory (Springer, New York, 2004)CrossRefMATH
15.
G. Cantor, Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. J. Reine Angew. Math. LXXVII, 258–263 (1873)
16.
17.
G. Cantor, Ludwig Scheffer (1859–1885). Necrolog. Bibl. Math. 197–199 (1885)
18.
G. Cantor, Ueber eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung 1, 75–78 (1892)MATH
19.
J. Cavaillès, E. Noether, Briefwechsel Cantor-Dedekind. Actualités Scientifiques et Industrielles, vol. 518 (Hermann, Paris, 1937)
20.
21.
O. Deiser, Relle Zahlen (Springer, Berlin, 2007)
22.
23.
24.
25.
26.
27.
28.
P. Erdős, Some personal and mathematical reminiscences of Kurt Mahler. Aust. Math. Soc. Gaz. 16, 1–2 (1989)MathSciNetMATH
29.
L. Euler, Introduction to analysis of the infinite. Book I, translated by J.D. Blanton (Springer, New York, 1988)
30.
G. Faber, Über stetige Funktionen. (Zweite Abhandlung). Math. Ann. 69, 372–443 (1910)
31.
H. Gispert, La théorie des ensembles en France avant la crise de 1905: Baire, Borel, Lebesgue … et tous les autres. Rev. Hist. Math. 1, 39–81 (1995)MathSciNet
32.
D. Gravé, Sur les lignes composées de parties rectilignes. C. R. Acad. Sci. Paris 127, 1005–1007 (1898)
33.
34.
J. Gray, Plato’s Ghost (Princeton University Press, Princeton, 2008)MATH
35.
36.
A. Harnack, Die Elemente der Differential- und Integralrechnung. Zur Einführung in das Studium dargestellt (Leipzig, Teubner, 1881)
37.
S. Hartman, Sur une condition supplémentaire dans les approximations diophantiques. Colloq. Math. 2, 48–51 (1949)MATH
38.
F. Hausdorff, Grundzüge der Mengenlehre (Veit & Co, Leipzig, 1914)MATH
39.
C. Hermite, Sur la fonction exponentielle. C. R. Acad. Sci. Paris 77, 18–24, 74–79, 226–233, 285–293 (1873)
40.
41.
I.N. Herstein, I. Kaplansky, Matters Mathematical (Harper & Row, New York, 1974); reprinted by Chelsea 1978
42.
43.
44.
A. Hurwitz, Ueber die Entwickelung der allgemeinen Theorie der analytischen Functionen in neuerer Zeit. Verh. Intern. Math. Congr. 1, 91–112 (1897)
45.
H.J. Ilgauds, W. Purkert, Georg Cantor (Birkhäuser, Basel, 1987)MATH
46.
47.
48.
A. Kanamori, Set theory from Cantor to Cohen, in Sets and Extensions in the Twentieth Century, ed. by D.M. Gabbay et al. Handbook of the History of Logic, vol. 6 (Elsevier, Boston, 2012), pp. 1–71
49.
A.J. Kempner, On transcendental numbers. Am. M.S. Bull. 22, 284–285 (1916)
50.
E. Knobloch, Beyond Cartesian limits: Leibniz’s passage from algebraic to ‘transcendental’ mathematics. Hist. Math. 33, 113–131 (2006)MathSciNetCrossRefMATH
51.
J.F. Koksma, Sur l’approximation des nombres irrationnels sous une condition supplementaire. Simon Stevin 28, 199–202 (1951)MathSciNetMATH
52.
L. Kronecker, Die Periodensysteme von Funktionen reeller Variablen. Ber. d. K. Preuß. Ak. d. Wiss. Berl. 1071–1080 (1884)
53.
D. Laugwitz, Debates about infinity in mathematics around 1890: the Cantor-Veronese controversy, its origins and its outcome. NTM (N.S.) 10, 102–126 (2002)
54.
J. Liouville, Remarques relatives 1∘ à des classes très-étendues de quantités dont la valeur n’est ni rationelle ni même réductible à des irrationnelles algébriques; 2∘ à un passage du livre des Principes où Newton calcule l’action exercée par une sphére sur un point extérieur. C.R. Acad. Sci. Paris 18, 883–885 (1844)
55.
J. Liouville, Nouvelle démonstration d’un théorème sur les irrationnelles algébriques. C.R. Acad. Sci. Paris 18, 910–911 (1844)
56.
J. Liouville, Sur des classes très-étendus de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationelles algébrique. Liouville J. 16, 133–142 (1851)
57.
J. Lützen, Joseph Liouville, 1809–1882: Master of Pure and Applied Mathematics (Springer, Berlin, 1990)CrossRefMATH
58.
59.
60.
S. Mazurkiewicz, Sur les fonctions non derivables. Stud. Math. 3, 92–94 (1931)MATH
61.
62.
G.H. Moore, Zermelo’s Axiom of Choice: Its Origins, Development, and Influence (Springer, New York, 1982)CrossRefMATH
63.
L. Olsen, Applications of multifractal divergence points to sets of numbers defined by their N-adic expansion. Math. Proc. Camb. Philos. Soc. 136, 139–165 (2004)MathSciNetCrossRefMATH
64.
W.F. Osgood, Ueber die ungleichmässige Convergenz und die gliedweise Integration der Reihen. Gött. Nachr. 288–291 (1896)
65.
66.
W.F. Osgood, On the existence of the Green’s function for the most general simply connected plane region. Trans. A.M.S. 1, 310–314 (1900)
67.
68.
69.
L. Scheeffer, Zur Theorie der stetigen Funktionen einer reellen Veränderlichen. Acta Math. 5, 183–194, 279–296 (1884)MathSciNetCrossRef
70.
T. Schneider, Einführung in die Transzendenten Zahlen, 2nd edn. (Springer, New York, 1957)CrossRefMATH
71.
72.
73.
74.
75.
W. Sierpiński, Sur la dualité entre la première categorie et la mesure nulle. Fund. Math. 22, 276–280 (1934)MATH
76.
W. Sierpiński, General Topology (University of Toronto Press, Toronto, 1952)MATH
77.
78.
H. Steinhaus, Anwendungen der Funktionalanalysis auf einige Fragen der reellen Funktionentheorie. Stud. Math. 1, 51–81 (1929)MATH
79.
E.B. van Vleck, A proof of some theorems on pointwise discontinuous functions. Trans. A.M.S. 8, 189–204 (1907)
80.
E.B. van Vleck, On non-measurable sets of points, with an example. Trans. A.M.S. 9, 237–244 (1908)
81.
E.B. van Vleck, Current tendencies of mathematical research. Bull. A.M.S. 23, 1–13 (1916)
82.
K. Volkert, Die Geschichte der pathologischen Funktionen. Ein Beitrag zur Entstehung der mathematischen Methodologie. Arch. Hist. Exact Sci. 37, 193–232 (1987)MathSciNetCrossRefMATH
83.
V. Volterra, Alcune osservazioni sulle funzioni punteggiate discontinue. Giornale di Matematiche 19, 76–86 (1881)
84.
85.
Metadata
Title
On Liouville Numbers: Yet Another Application of Functional Analysis to Number Theory
Author
Jörn Steuding
Copyright Year
2016
Book
From Arithmetic to Zeta-Functions

Print ISBN: 978-3-319-28202-2

Electronic ISBN: 978-3-319-28203-9

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-28203-9_29

Premium Partner

    Image Credits