Abstract
We study 0-1-sequences and establish the connection between the values of the upper and lower Sucheston functional on such sequence and the set of all possible divisors of the elements in the sequence support. If the union of the sets of all simple divisors of the elements in a 0-1-sequence support is finite then the sequence is almost convergent to zero. We study the 0-1-sequences whose support consists exactly of the multiples of the elements in a given set, and establish some necessary and sufficient conditions for the upper Sucheston functional to be equal to 1 on such sequence. We prove that there are infinitely many sequences at which the lower Sucheston functional is 1, and the lower Sucheston functional never vanishes at any of such sequences.
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References
Mazur S., “O metodach sumowalności,” in: Ksiega Pamiatkowa Pierwszego Polskiego Zjazdu Matematycznego, Lwow, 7–10.IX.1927. Dodatek do Annales de la Société Polonaise de Mathématique (2013), 102–107 (Polskie Towarzystwo Matematyczne). https://www.ptm.org.pl/zjazd/pliki/102-107.html
Banach S., Théorie des Opérations Linéares, Mathematisches Seminar der Univ. Warschau, Warszawa (1932) (Monografje Matematyczne I).
Lorentz G.G., “A contribution to the theory of divergent sequences,” Acta Math., vol. 80, no. 1, 167–190 (1948).
Sucheston L., “Banach limits,” Amer. Math. Monthly, vol. 74, no. 3, 308–311 (1967).
Semenov E.M., Sukochev F.A., and Usachev A.S., “Geometry of Banach limits and their applications,” Russian Math. Surveys, vol. 75, no. 4, 725–763 (2020).
Avdeev N.N., “On the space of almost convergent sequences,” Math. Notes, vol. 105, no. 3, 462–466 (2019).
Hall R.R. and Tenenbaum G., “On Behrend sequences,” in: Mathematical Proceedings of the Cambridge Philosophical Society. vol. 112, Cambridge University, Cambridge (1992), 467–482.
Davenport H. and Erdos P., “On sequences of positive integers,” Acta Arithm., vol. 2, 147–151 (1936).
Davenport H. and Erdos P., “On sequences of positive integers,” J. Indian Math. Soc., N. S., vol. 15, 19–24 (1951).
Besicovitch A., “On the density of certain sequences of integers,” Math. Ann., vol. 100, no. 1, 336–341 (1935).
Hall R.R., Sets of Multiples, Cambridge University, Cambridge (1996) (Cambridge Tracts in Mathematics).
Euler L., “Variae observationes circa series infinitas,” in: The Early Mathematics of Leonhard Euler. The MAA Tercentenary Euler Celebration. AMS/MAA Spectrum, vol. 98, MAA, Providence (2007), 249–260 (Commentarii Academiae Scientiarum Imperialis Petropolitanae, vol. 9, 160–168 (1737)). https://www.biodiversitylibrary.org/page/10093417
Funding
The work was supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS” (Project 22–7–2–27–2) and the Russian Science Foundation (Project 19–11–00197).
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Translated from Vladikavkazskii Matematicheskii Zhurnal, 2021, Vol. 23, No. 4, pp. 5–14. https://doi.org/10.46698/p9825-1385-3019-c
To the blessed memory of Professor Aleksandr Dmitrievich Baev, Dean of the Mathematical Faculty of Voronezh State University.
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Avdeev, N.N. Almost Convergent 0-1-Sequences and Primes. Sib Math J 64, 1455–1461 (2023). https://doi.org/10.1134/S0037446623060174
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DOI: https://doi.org/10.1134/S0037446623060174
Keywords
- space of bounded sequences
- Banach limit
- Sucheston functional
- almost convergent sequence
- 0-1-sequence
- integer factorization
- subsets of naturals