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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2406.09114 |
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| _version_ | 1866917692892512256 |
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| author | Weiß, Christian |
| author_facet | Weiß, Christian |
| contents | The classic example of a low-discrepancy sequence in $\mathbb{Z}_p$ is $(x_n) = an+b$ with $a \in \mathbb{Z}_p^x$ and $b \in \mathbb{Z}_p$. Here we address the non-linear case and show that a polynomial $f$ generates a low-discrepancy sequence in $\mathbb{Z}_p$ if and only if it is a permutation polynomial $\mod p$ and $\mod p^2$. By this it is possible to construct non-linear examples of low-discrepancy sequences in $\mathbb{Z}_p$ for all primes $p$. Moreover, we prove a criterion which decides for any given polynomial in $\mathbb{Z}_p$ with $p \in \left\{ 3,5, 7\right\}$ if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial p-adic Low-Discrepancy Sequences Weiß, Christian Number Theory The classic example of a low-discrepancy sequence in $\mathbb{Z}_p$ is $(x_n) = an+b$ with $a \in \mathbb{Z}_p^x$ and $b \in \mathbb{Z}_p$. Here we address the non-linear case and show that a polynomial $f$ generates a low-discrepancy sequence in $\mathbb{Z}_p$ if and only if it is a permutation polynomial $\mod p$ and $\mod p^2$. By this it is possible to construct non-linear examples of low-discrepancy sequences in $\mathbb{Z}_p$ for all primes $p$. Moreover, we prove a criterion which decides for any given polynomial in $\mathbb{Z}_p$ with $p \in \left\{ 3,5, 7\right\}$ if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy. |
| title | Polynomial p-adic Low-Discrepancy Sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.09114 |