Degree of independence of numbers
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Jittinart Rattanamoong
Abstract
A new concept of independence of real numbers, called degree independence, which contains those of linear and algebraic independences, is introduced. A sufficient criterion for such independence is established based on a 1988 result of Bundschuh, which in turn makes use of a generalization of Liouville’s estimate due to Feldman in 1968. Applications to numbers represented by Cantor series and product expansions are derived.
Communicated by István Gaál
References
[1] Bugeau, Y.: Approximation by Algebraic Numbers, Cambridge University Press, Cambridge, 2004.10.1017/CBO9780511542886Search in Google Scholar
[2] Bundschuh, P.: A criterion for algebraic independence with some applications, Osaka J. Math. 25 (1988), 849–858.Search in Google Scholar
[3] Fel’dman, N. I.: Estimate for a linear form of logarithms of algebraic numbers, Mat. Sb. (N.S.) 76 (1968), 304-319 (in Russian); Engl. transl.: Math. USSR-Sb. 5 (1968), 291–307.Search in Google Scholar
[4] Komatsu, T. et al.: Complex Cantor series, Cantor products and their independence, Integers 19 (2019), ♯A57.Search in Google Scholar
[5] LeVeque, W. J.: Topics in Number Theory, Volume II, Addison-Wesley, Reading, 1956.Search in Google Scholar
[6] Liouville, J.: Sur des classes très-étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques, J. Math. pures appl. 16 (1851), 133–142.Search in Google Scholar
[7] von Neumann, J.: Ein System algebraisch unabhängiger Zahlen, Math. Ann. 99 (1928), 134–141.10.1007/BF01459089Search in Google Scholar
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Articles in the same Issue
- Regular papers
- Properties of implication in effect algebras
- On a nonlinear relation for computing the overpartition function
- Six-cycle systems
- Representation of bifinite domains by BF-closure spaces
- L-fuzzy cosets in universal algebras
- Density of sets with missing differences and applications
- Degree of independence of numbers
- A generalization of a result on the sum of element orders of a finite group
- A New fuzzy McShane integrability
- Hankel determinants of second and third order for the class 𝓢 of univalent functions
- Herglotz's theorem for Jacobi-Dunkl positive definite sequences
- Functional inequalities for Gaussian hypergeometric function and generalized elliptic integral of the first kind
- Existence on solutions of a class of casual differential equations on a time scale
- On a general system of difference equations defined by homogeneous functions
- Jordan amenability of banach algebras
- A new notion of orthogonality involving area and length
- Reeb flow invariant ∗-Ricci operators on trans-Sasakian three-manifolds
- A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function
- On first countable quasitopological homotopy groups
