Measures associated with certain ellipsephic harmonic series and the Allouche-Hu-Morin limit theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917147804958720 |
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| author | Burnol, Jean-François |
| author_facet | Burnol, Jean-François |
| contents | We consider the harmonic series $S(k)=\sum^{(k)} m^{-1}$ over the integers having $k$ occurrences of a given block of $b$-ary digits, of length $p$, and relate them to certain measures on the interval $[0,1)$. We show that these measures converge weakly to $b^p$ times the Lebesgue measure, a fact which allows a new proof of the theorem of Allouche, Hu, and Morin which says $\lim S(k)=b^p\log(b)$. A quantitative error estimate will be given. Combinatorial aspects involve generating series which fall under the scope of the Goulden-Jackson cluster generating function formalism and the work of Guibas-Odlyzko on string overlaps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_03625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measures associated with certain ellipsephic harmonic series and the Allouche-Hu-Morin limit theorem Burnol, Jean-François Number Theory Combinatorics 11Y60, 05A15 (Primary) 11A63, 28A25 (Secondary) We consider the harmonic series $S(k)=\sum^{(k)} m^{-1}$ over the integers having $k$ occurrences of a given block of $b$-ary digits, of length $p$, and relate them to certain measures on the interval $[0,1)$. We show that these measures converge weakly to $b^p$ times the Lebesgue measure, a fact which allows a new proof of the theorem of Allouche, Hu, and Morin which says $\lim S(k)=b^p\log(b)$. A quantitative error estimate will be given. Combinatorial aspects involve generating series which fall under the scope of the Goulden-Jackson cluster generating function formalism and the work of Guibas-Odlyzko on string overlaps. |
| title | Measures associated with certain ellipsephic harmonic series and the Allouche-Hu-Morin limit theorem |
| topic | Number Theory Combinatorics 11Y60, 05A15 (Primary) 11A63, 28A25 (Secondary) |
| url | https://arxiv.org/abs/2405.03625 |