Uniform distribution via lattices: from point sets to sequences
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912240517513216 |
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| author | Ferizović, Damir |
| author_facet | Ferizović, Damir |
| contents | In this work we construct many sequences $S=S^\Box_{b,d}$, or $S=S^\boxplus_{b,d}$ in the $d$--dimensional unit hypercube, which for $d=1$ are (generalized) van der Corput sequences or Niederreiter's $(0,1)$-sequences in base $b$ respectively. Further, we introduce the notion of $f$-sublinearity and use it to define discrepancy functions which subsume the notion of $L^p$-discrepancy, Wasserstein $p$-distance, and many more methods to compare empirical measures to an underlying base measure.
We will relate bounds for a given discrepancy functions $\mathscr{D}$ of the multiset of projected lattice sets $P(b^{-m}\mathbb{Z}^d$), to bounds of $\mathscr{D}(Z_N)$, i.e. the initial segments of the sequence $Z=P(S)$ for any $N\in\mathbb{N}$. We show that this relation holds in any dimension $d$, for any map $P$ defined on a hypercube, and any discrepancy function as introduced in this work for which bounds on $P(b^{-m}\mathbb{Z}^d+v$) can be obtained.
We apply this theorem in $d=1$ to obtain bounds for the $L^p$--discrepancy of van der Corput and Niederreiter (0,1) sequences in terms of digit sums for all $0<p\leq \infty$. In $d=2$ an application of our construction yields many sequences on the two-sphere, such that the initial segments $Z_N$ have low $L^\infty$--discrepancy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_13297 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform distribution via lattices: from point sets to sequences Ferizović, Damir Classical Analysis and ODEs 11K36 (primary), 11K38, 52C99 In this work we construct many sequences $S=S^\Box_{b,d}$, or $S=S^\boxplus_{b,d}$ in the $d$--dimensional unit hypercube, which for $d=1$ are (generalized) van der Corput sequences or Niederreiter's $(0,1)$-sequences in base $b$ respectively. Further, we introduce the notion of $f$-sublinearity and use it to define discrepancy functions which subsume the notion of $L^p$-discrepancy, Wasserstein $p$-distance, and many more methods to compare empirical measures to an underlying base measure. We will relate bounds for a given discrepancy functions $\mathscr{D}$ of the multiset of projected lattice sets $P(b^{-m}\mathbb{Z}^d$), to bounds of $\mathscr{D}(Z_N)$, i.e. the initial segments of the sequence $Z=P(S)$ for any $N\in\mathbb{N}$. We show that this relation holds in any dimension $d$, for any map $P$ defined on a hypercube, and any discrepancy function as introduced in this work for which bounds on $P(b^{-m}\mathbb{Z}^d+v$) can be obtained. We apply this theorem in $d=1$ to obtain bounds for the $L^p$--discrepancy of van der Corput and Niederreiter (0,1) sequences in terms of digit sums for all $0<p\leq \infty$. In $d=2$ an application of our construction yields many sequences on the two-sphere, such that the initial segments $Z_N$ have low $L^\infty$--discrepancy. |
| title | Uniform distribution via lattices: from point sets to sequences |
| topic | Classical Analysis and ODEs 11K36 (primary), 11K38, 52C99 |
| url | https://arxiv.org/abs/2308.13297 |