Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918233918930944 |
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| author | Fan, Aihua Fan, Shilei Ye, Hanfei |
| author_facet | Fan, Aihua Fan, Shilei Ye, Hanfei |
| contents | We establish a non-Archimedean analogue of Koksma's theorem. For a local field F of characteristic zero, we prove that the sequence ([αx^n]) is uniformly distributed in the valuation ring O for almost every x with |x|_p>1. In the case of positive characteristic, ([x^n]) fails to be uniformly distributed, but it becomes μ*-uniformly distributed for some weighted measure μ*. These results are derived from a general metric theorem for sequences generated by expanding scaling maps. On the other hand, we demonstrate that the exceptional set of parameters x for which these sequences are not uniformly distributed is large (i.e. having full Hausdorff dimension) and share a rich q-homogeneous fractal structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_05690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets Fan, Aihua Fan, Shilei Ye, Hanfei Number Theory Classical Analysis and ODEs Dynamical Systems We establish a non-Archimedean analogue of Koksma's theorem. For a local field F of characteristic zero, we prove that the sequence ([αx^n]) is uniformly distributed in the valuation ring O for almost every x with |x|_p>1. In the case of positive characteristic, ([x^n]) fails to be uniformly distributed, but it becomes μ*-uniformly distributed for some weighted measure μ*. These results are derived from a general metric theorem for sequences generated by expanding scaling maps. On the other hand, we demonstrate that the exceptional set of parameters x for which these sequences are not uniformly distributed is large (i.e. having full Hausdorff dimension) and share a rich q-homogeneous fractal structure. |
| title | Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets |
| topic | Number Theory Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2512.05690 |