The digits of the Pierce expansion satisfy the law of large numbers. It is known that the Hausdorff dimension of the set of exceptions to the law of large numbers is $ 1 $. We provide an elementary proof of this fact by adapting Jun Wu's method, which was originally used for Engel expansions. Our approach emphasizes the fractal nature of exceptional sets and avoids advanced machinery, thereby relying instead on explicit sequences and constructive techniques. Furthermore, our method opens the possibility of extending similar analyses to other real number representation systems, such as the Engel, Lüroth, and Sylvester expansions, thus paving the way for further explorations in metric number theory and fractal geometry.
Citation: Min Woong Ahn. An elementary proof that the set of exceptions to the law of large numbers in Pierce expansions has full Hausdorff dimension[J]. AIMS Mathematics, 2025, 10(3): 6025-6039. doi: 10.3934/math.2025275
The digits of the Pierce expansion satisfy the law of large numbers. It is known that the Hausdorff dimension of the set of exceptions to the law of large numbers is $ 1 $. We provide an elementary proof of this fact by adapting Jun Wu's method, which was originally used for Engel expansions. Our approach emphasizes the fractal nature of exceptional sets and avoids advanced machinery, thereby relying instead on explicit sequences and constructive techniques. Furthermore, our method opens the possibility of extending similar analyses to other real number representation systems, such as the Engel, Lüroth, and Sylvester expansions, thus paving the way for further explorations in metric number theory and fractal geometry.
| [1] |
M. W. Ahn, On the error-sum function of Pierce expansions, J. Fractal Geom., 10 (2023), 389–421. http://dx.doi.org/10.4171/JFG/142 doi: 10.4171/JFG/142
|
| [2] |
J. O. Shallit, Metric theory of Pierce expansions, Fibonacci Quart., 24 (1986), 22–40. https://doi.org/10.1080/00150517.1986.12429786 doi: 10.1080/00150517.1986.12429786
|
| [3] | J. Galambos, Representations of real numbers by infinite series, Berlin, Heidelberg: Springer, 1976. https://doi.org/10.1007/BFb0081642 |
| [4] | P. Erdős, A. Rényi, P. Szüsz, On Engel's and Sylvester's series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 1 (1958), 7–32. |
| [5] |
J. Wu, A problem of Galambos on Engel expansions, Acta Arith., 92 (2000), 383–386. https://doi.org/10.4064/aa-92-4-383-386 doi: 10.4064/aa-92-4-383-386
|
| [6] |
M. W. Ahn, Hausdorff dimensions in Pierce expansions, Acta Arith., 215 (2024), 115–160. https://doi.org/10.4064/aa230427-18-3 doi: 10.4064/aa230427-18-3
|
| [7] | K. Falconer, Fractal geometry: Mathematical foundations and applications, 3 Eds., Chichester: Wiley, 2014. |
| [8] |
M. W. Ahn, Exceptional sets to Shallit's law of leap years in Pierce expansions, J. Math. Anal. Appl., 545 (2025), 129124. https://doi.org/10.1016/j.jmaa.2024.129124 doi: 10.1016/j.jmaa.2024.129124
|
| [9] |
J. O. Shallit, Pierce expansions and rules for the determination of leap years, Fibonacci Quart., 32 (1994), 416–423. https://doi.org/10.1080/00150517.1994.12429190 doi: 10.1080/00150517.1994.12429190
|
Min Woong Ahn. An elementary proof that the set of exceptions to the law of large numbers in Pierce expansions has full Hausdorff dimension[J]. AIMS Mathematics, 2025, 10(3): 6025-6039. doi: 10.3934/math.2025275
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