Signed null sequences and Hausdorff dimension
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914053987762176 |
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| author | Balka, Richárd Héra, Kornélia Kiss, Gergely |
| author_facet | Balka, Richárd Héra, Kornélia Kiss, Gergely |
| contents | We investigate the convergence of signed null sequences of the form \[ \sum_{n=1}^\infty \varepsilon_n a_n, \quad \varepsilon_n \in \{-1,1\}, \] where $(a_n)$ tends to zero in $\mathbb{R}^d$. Our main result shows that for any such sequence, the set of sign sequences yielding convergence has full Hausdorff dimension in the natural ultrametric topology. This answers a question of Mattila in the one-dimensional case, for which we provide an elementary proof. Moreover, if $(a_n)\notin \ell^1$ in one dimension, then for every $L\in\mathbb{R}$ the set of sign sequences with sum $L$ also has Hausdorff dimension $1$. In higher dimensions the analogous statement does not hold in full generality, but it is guaranteed if the sequence has $d$ linearly independent Lévy vectors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_20181 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Signed null sequences and Hausdorff dimension Balka, Richárd Héra, Kornélia Kiss, Gergely Classical Analysis and ODEs 28A78, 28A80, 51F30, 54E45 We investigate the convergence of signed null sequences of the form \[ \sum_{n=1}^\infty \varepsilon_n a_n, \quad \varepsilon_n \in \{-1,1\}, \] where $(a_n)$ tends to zero in $\mathbb{R}^d$. Our main result shows that for any such sequence, the set of sign sequences yielding convergence has full Hausdorff dimension in the natural ultrametric topology. This answers a question of Mattila in the one-dimensional case, for which we provide an elementary proof. Moreover, if $(a_n)\notin \ell^1$ in one dimension, then for every $L\in\mathbb{R}$ the set of sign sequences with sum $L$ also has Hausdorff dimension $1$. In higher dimensions the analogous statement does not hold in full generality, but it is guaranteed if the sequence has $d$ linearly independent Lévy vectors. |
| title | Signed null sequences and Hausdorff dimension |
| topic | Classical Analysis and ODEs 28A78, 28A80, 51F30, 54E45 |
| url | https://arxiv.org/abs/2509.20181 |