### Abstract

The main purpose of this paper is to discuss the asymptotic behaviour of the difference s_{q,k} (P(n)) - k(q - 1)/2 where s_{q,k}(n) denotes the sum of the first k digits in the q-ary digital expansion of n and P(x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P, a Strassen type version of the law of the iterated logarithm. Furthermore, we extend these results to the joint distribution of q_{1}-ary and q_{2}-ary digital expansions where q_{1} and q_{2} are coprime.

Original language | English |
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Pages (from-to) | 175-201 |

Number of pages | 27 |

Journal | Acta Mathematica Hungarica |

Volume | 98 |

Issue number | 3 |

DOIs | |

State | Published - 1 Feb 2002 |

### Keywords

- Functional limit theorem
- Q-ary digital expansion
- Strassen's law of the iterated logarithm
- Sum-of-digits function

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## Cite this

Drmota, M., Fuchs, M., & Manstavičius, E. (2002). Functional limit theorems for digital expansions.

*Acta Mathematica Hungarica*,*98*(3), 175-201. https://doi.org/10.1023/A:1022869708089