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A note on exceptional sets in Erdös–Rényi limit theorem

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Abstract

For \(x\in [0,1]\), the run-length function \(r_n(x)\) is defined as the length of longest run of 1’s among the first n dyadic digits in the dyadic expansion of x. We study the Hausdorff dimension of the exceptional set in Erdös–Rényi limit theorem. Let \(\varphi :{{\mathbb {N}}}\rightarrow (0,\infty )\) be a monotonically increasing function with \(\lim _{n\rightarrow \infty }\varphi (n)=\infty \) and \(0\le \alpha \le \beta \le \infty \), define

$$\begin{aligned} E_{\alpha ,\beta }^\varphi =\Big \{x\in [0,1]:\, \liminf _{n\rightarrow \infty } \dfrac{r_n(x)}{\varphi (n)}=\alpha , \limsup _{n\rightarrow \infty } \frac{r_n(x)}{\varphi (n)}=\beta \Big \}. \end{aligned}$$

We prove that \(E_{\alpha ,\beta }^\varphi \) has Hausdorff dimension one if \(\lim _{n,p\rightarrow \infty }\frac{\varphi (n+p)-\varphi (n)}{p}=0\) and that \(E_{0,\infty }^\varphi \) is residual in [0,1] when \(\liminf _{n\rightarrow {\infty }}\frac{\varphi (n)}{n}=0\).

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References

  1. Albeverio, S., Pratsiovytyi, M., Torbin, G.: Topological and fractal properties of subsets of real numbers which are not normal. Bull. Sci. Math. 129, 615–630 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, H.B., Wen, Z.X.: The fractional dimensions of intersections of the Besicovitch sets and the Erdös-Rényi sets. J. Math. Anal. Appl. 401, 29–37 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Erdös, P., Rényi, A.: On a new law of large numbers. J. Anal. Math. 22, 103–111 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fan, A.H., Feng, D.J., Wu, J.: Recurrence, dimensions and entropies. J. Lond. Math. Soc. 64, 229–244 (2001)

    Article  Google Scholar 

  5. Fang, L.L., Ma, J.H., Song, K.K., Wu, M.: Exceptional sets related to the run-length function of beta-expansions. Fractals 26(4), 1850049, 10 pp (2018)

  6. Feng, D.J., Wu, J.: The Hausdorff dimension of recurrent sets in symbolic spaces. Nonlinearity 14, 81–85 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hyde, J., Laschos, V., Olsen, L., Petrykiewicz, I., Shaw, A.: Iterated Cesàro averages, frequencies of digits and Baire category. Acta Arith. 144, 287–293 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, J.J., Wu, M.: The sets of divergence points of self-similar measures are residual. J. Math. Anal. Appl. 404, 429–437 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, J.J., Wu, M.: On exceptional sets in Erdös-Rényi limit theorem. J. Math. Anal. Appl. 436, 355–365 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, J.J., Wu, M.: On exceptional sets in Erdös-Rényi limit theorem revisited. Monatsh. Math. 182(4), 865–875 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, C.T., Li, H.X.: Hausdorff dimension of local level sets of Takagi’s function. Monatsh. Math. 177(1), 101–117 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, C.T.: Some topological results on level sets and multifractal sets. Fractals 28(01), 2050009 (2020)

    Article  MATH  Google Scholar 

  13. Liu, J., Lü, M.Y., Zhang, Z.L.: On the exceptional sets in Erdös-Rényi limit theorem of \(\beta \) -expansion. Int. J. Number Theory 14(7), 1919–1934 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lüroth, J.: Über eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe. Math. Ann. 21, 411–423 (1883). (In German)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ma, J.H., Wen, S.Y., Wen, Z.Y.: Egoroff’s theorem and maximal run length. Monatsh. Math. 151, 287–292 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Olsen, L.: Extremely non-normal numbers. Math. Proc. Cambridge Philos. Soc. 137, 43–53 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Oxtoby, J.C.: Measure and Category. Springer, New York (1996)

    Google Scholar 

  18. Sun, Y., Xu, J.: A remark on exceptional sets in Erdös-Rényi limit theorem. Monatsh. Math. 184, 291–296 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, Y., Xu, J.: On the maximal run-length function in the Lüroth expansion. Czech. Math. J. 68(1), 277–291 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous referee for many valuable suggestions. The author is supported by the National Natural Science Foundation of China (11601403) and Hubei Provincial Natural Science Foundation of China (2019CFB602).

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  1. College of Mathematics and Physics, Guangxi Minzu University, Nanning, 530006, People’s Republic of China

    Chuntai Liu

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  1. Chuntai Liu
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Correspondence to Chuntai Liu.

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Communicated by Constantin Niculescu.

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Liu, C. A note on exceptional sets in Erdös–Rényi limit theorem. Ann. Funct. Anal. 14, 71 (2023). https://doi.org/10.1007/s43034-023-00294-w

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