On representation by continued fractions of elements of some special algebras

S. M. Khryashchev

Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part III, Tome 539 (2024), pp. 180-213

Résumé

In this paper we consider continued fractions with elements from some special algebras. In absence of commutativity under multiplication, the study of properties of continued fractions is significantly complicated. For such fractions, some concepts related to classical continued fractions need to be modified. However, on the other hand, many properties of classical continued fractions are fulfilled for the generalized continued fractions under consideration. In particular, the values of the residuals are calculated using similar formulas. The most complete results can be obtained for continued fractions on the set of quaternions.

Bibliographie

[1] A. Ya. Khinchin, Tsepnye drobi, Gos. izd. fiz.-mat. lit, M., 1961 | MR

[2] A. A. Bukhshtab, Teoriya chisel, Izd-vo Prosveschenie, M., 1966, 385 pp.

[3] S. Leng, Vvedenie v teoriyu diofantovykh priblizhenii, Mir, M., 1966 | MR

[4] S. G. Dani, Arnaldo Nogueira, “Continued fractions for complex numbers and values of binary quadratic forms”, Transactions of the American Mathematical Society, 366:7, 3553–3583 | DOI | MR | Zbl

[5] S. G. Dani, “Continued fraction expansions for complex numbers”, General Approach. Acta Arith., 171 (2015), 355–369 | DOI | MR | Zbl

[6] S. G. Dani, Ojas Sahasrabudhe, Generalized continued fraction expansions of complex numbers, and applications to quadratic and badly approximable numbers, 2021, arXiv: 2102.11769v1 [math. NT]

[7] V. I. Arnold, Tsepnye drobi, Izd-vo MTsNMO, M., 2009

[8] A. D. Bruno, “On generalizations of the continued fraction”, Preprinty of the Keldysh Institute of Applied Mathematics of RAS, 2004 | MR

[9] V. I. Parusnikov, “Chetyrekhmernoe obobschenie tsepnykh drobei”, Preprinty Instituta prikladnoi matematiki im. M. V. Keldysha RAN, 2011 | Zbl

[10] A. N. Korobov, “Mnogomernye tsepnye drobi i otsenki lineinykh form”, Acta Arithmetica, LXXI:4 (1995) | DOI | MR | Zbl

[11] A. A. Lodkin, “Parus Kleina i diofantovy priblizheniya vektora”, Zap. nauchn. semin. POMI, 481, 2019, 63–73

[12] Yusuf Hartono, Ergodic Properties of Continued Fraction Algorithms, DUP Science is an imprint of Delft University Press, Delft, The Netherlands, 2003 | MR

[13] S. M. Khryashchev, “On Finding Switching Instants for Control of Discrete-time Dynamical Polysystems by Using Continued Fractions”, Proceedings of 15th International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiys Conference), STAB 2020, 2020, 9140678 | DOI

[14] S. M. Khryashchev, “A Method of Determining of Switching Instants for Discrete-time Control Systems”, Stability and Control Processes, SCP 2020, Proceedings, Lecture Notes in Control and Information Sciences, Springer, Cham, 2020 | MR

[15] S. M. Khryashchev, “A Method of Finding Discrete Controls for Switched Dynamical Systems by Applying Continued Fractions”, Nonlinear Phenomena in Complex Systems, 24:2 (2021), 175–183 | DOI | Zbl

[16] S. M. Khryashchev, “On the Application of Multicomponent Continued Fractions to Control Problems for Dynamical Systems”, Proceedings of 16th International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiys Conference), STAB 2022, 2022, 9140678 | DOI

[17] S. M. Khryashchev, “On Multicomponent Continued Fraction Expansions of Hypernumbers of Certain Classes”, Proceedings of International Conference on Polynomial Computer Algebra, St. Petersburg, 2024, 71–75

[18] Leng S., Algebra, Mir, M., 1968

[19] B. L. van der Varden, Algebra, Lan, SPb., 2004 | MR

[20] N. G. Marchuk, Uravneniya teorii polya i algebry Klifforda, NITs “Regulyarnaya i khaoticheskaya dinamika”, M.-Izhevsk, 2009