Multiplicative group generated by quotients of integral parts (original) (raw)

Abstract

For fixed positive numbers \(\alpha \ne \beta \), we consider the multiplicative group \({\mathcal {F}}_{\alpha ,\beta }\) generated by the rational numbers of the form \(\frac{\lfloor \alpha n\rfloor }{\lfloor \beta n\rfloor }\), where \(n \in {\mathbb {N}}\) and \(n \ge \max \big (\frac{1}{\alpha },\frac{1}{\beta }\big )\). For \(0<\alpha <1\) and \(\beta =1\), we prove that \({\mathcal {F}}_{\alpha ,1}\) is the maximal possible group, namely, it is the multiplicative group of all positive rational numbers \({\mathbb {Q}}^{+}\). The same equality \({\mathcal {F}}_{\alpha ,\beta }={\mathbb {Q}}^{+}\) holds in the case when \(0< 10 \alpha \le \beta <1\). These results produce infinitely many pairs \((\alpha ,\beta )\), \(\alpha \ne \beta \), for which a conjecture posed by Kátai and Phong can be confirmed. The only previously known such example is \((\alpha ,\beta )=(\sqrt{2},1)\). We also give a new proof in that case, which is much simpler than the original proof of Kátai and Phong. More generally, we conjecture that \({\mathcal {F}}_{\alpha ,\beta }={\mathbb {Q}}^{+}\) for \(\alpha \ne \beta \) if at least one of the numbers \(\alpha ,\beta >0\) is not an integer.

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References

  1. J.M. De Koninck, I. Kátai, B.M. Phong, Almost-additive and almost-multiplicative functions with regularity properties. Publ. Math. Debrecen 99, 151–160 (2021)
    Article MathSciNet Google Scholar
  2. A. Hildebrand, An Erdős–Wintner theorem for differences of additive functions. Trans. Am. Math. Soc. 310, 257–276 (1988)
    Google Scholar
  3. I. Kátai, Multiplicative functions with regularity properties. II. Acta Math. Hungar. 42, 295–308 (1983)
    Article MathSciNet Google Scholar
  4. I. Kátai, B.M. Phong, Some unsolved problems on arithmetical functions. Ann. Univ. Sci. Budapest. Sect. Comput. 44, 233–235 (2015)
    MathSciNet Google Scholar
  5. I. Kátai, B.M. Phong, On the multiplicative group generated by \(\{\frac{[\sqrt{2}n]}{n}\mid n\in \mathbb{N} \}\). Acta Math. Hungar. 145, 80–87 (2015)
    Article MathSciNet Google Scholar
  6. I. Kátai, B.M. Phong, On the multiplicative group generated by \(\{\frac{[\sqrt{2}n]}{n}\mid n\in \mathbb{N} \}\). II. Acta Sci. Math. (Szeged) 81, 431–436 (2015)
    Article MathSciNet Google Scholar
  7. I. Kátai, B.M. Phong, On the multiplicative group generated by \(\{\frac{[\sqrt{2}n]}{n}\mid n\in \mathbb{N} \}\). III. Acta Math. Hungar. 147, 247–254 (2015)
    Article MathSciNet Google Scholar
  8. I. Kátai, B.M. Phong, On the multiplicative group generated by \(\{\frac{[\sqrt{2}n]}{n}\mid n\in \mathbb{N} \}\). IV. Math. Pannon. 25, 105–112 (2014/2015)
  9. I. Kátai, B.M. Phong, On the multiplicative group generated by \(\{\frac{[\sqrt{2}n]}{n}\mid n\in \mathbb{N} \}\). V. Notes Number Theory Discrete Math. 29, 348–353 (2023)
    Article Google Scholar
  10. O. Klurman, Correlations of multiplicative functions and applications. Compos. Math. 153, 1622–1657 (2017)
    Article MathSciNet Google Scholar
  11. O. Klurman, P.A. Mangerel, Rigidity theorems for multiplicative functions. Math. Ann. 372, 651–697 (2018)
    Article MathSciNet Google Scholar
  12. K.-L. Kueh, A note on Kronecker’s approximation theorem. Am. Math. Mon. 93, 555–556 (1986)
    Article MathSciNet Google Scholar
  13. P. Ribenboim, The New Book of Prime Number Records (Springer, New York, 1996)
    Book Google Scholar
  14. E. Wirsing, Y.-S. Tang, P.-T. Shao, On a conjecture of Kátai for additive functions. J. Number Theory 56, 391–395 (1996)
    Article MathSciNet Google Scholar

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  1. Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania
    Artūras Dubickas

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Correspondence toArtūras Dubickas.

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Dubickas, A. Multiplicative group generated by quotients of integral parts.Period Math Hung 89, 355–362 (2024). https://doi.org/10.1007/s10998-024-00599-w

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