Hankel Matrix Decompositions and Digital Sequences over Function Fields

Abstract

This paper studies formal Laurent series, with particular emphasis on their continued-fraction and Laurent-series expansions. Knowledge of both expansions is important, for example, in the study of Kronecker-type sequences in uniform distribution theory and in the search for potential counterexamples to variants of Littlewood's conjecture over fields of formal Laurent series. Our approach investigates Hankel matrices over a field and relates them to Laurent-series expansions over the same field. We describe such matrices via a block ℒℬ𝒰 decomposition, using recursive algorithms based on the continued-fraction coefficients of the corresponding Laurent series. The upper-triangular factor 𝒰 has several remarkable properties. In particular, it is constructed columnwise from the so-called Ostrowski-type representations of powers of X. Applying 𝒰 in Niederreiter's digital construction yields an Ostrowski-type van der Corput sequence that can be related to the corresponding Kronecker-type sequence. For ordinary Kronecker sequences, such a relation is known in only one case, that of the (nφ)-sequence. The original motivation for our study of Hankel matrices was to recast the search for counterexamples to variants of Littlewood's conjecture over fields of Laurent series as the search for specific Kronecker-type sequences. Interestingly, the method developed for the study of Kronecker-type sequences also allows a new perspective on the celebrated Sobol' construction. Our approach yields a short and elegant proof of its excellent distribution properties.

Hankel Transform and (α,β) Somos-4 Sequences
Feihu Liu, Ying Wang, Zihao Zhang
arXiv16 Oct 2026PreprintCC0Full textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
On the Unified Generalized q-Stirling Matrices and Their Matrix Decompositions
Amerah M. Dibagulun, Charles B. Montero, Mahid M. Mangontarum
arXiv15 Sep 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
Generalized Hankel/Toeplitz matrix for array signal processing
Wenchong Huang, Kunpeng Li, Ping Liu
arXiv3 Sep 2026PreprintarXiv licenceFull textGeneral mathematicsarXiv
The threshold for fractional clique decompositions of random hypergraphs, via matrix scaling
Tuan Tran
arXiv1 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematics ProbabilityarXiv
Coefficient estimates and Hankel determinant for the class of leaf shaped starlike function
Shantanu Panja, Abhijit Banerjee, Sujoy Majumder
arXiv1 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{hofer2026hankel,
  title = {{Hankel Matrix Decompositions and Digital Sequences over Function Fields}},
  author = {Roswitha Hofer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12691},
  eprint = {2610.12691},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12691},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Hankel Matrix Decompositions and Digital Sequences over Function Fields
AU  - Roswitha Hofer
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12691
UR  - https://arxiv.org/abs/2610.12691
AB  - This paper studies formal Laurent series, with particular emphasis on their continued-fraction and Laurent-series expansions. Knowledge of both expansions is important, for example, in the study of Kronecker-type sequences in uniform distribution theory and in the search for potential counterexamples to variants of Littlewood's conjecture over fields of formal Laurent series. Our approach investigates Hankel matrices over a field and relates them to Laurent-series expansions over the same field. We describe such matrices via a block ℒℬ𝒰 decomposition, using recursive algorithms based on the continued-fraction coefficients of the corresponding Laurent series. The upper-triangular factor 𝒰 has several remarkable properties. In particular, it is constructed columnwise from the so-called Ostrowski-type representations of powers of X. Applying 𝒰 in Niederreiter's digital construction yields an Ostrowski-type van der Corput sequence that can be related to the corresponding Kronecker-type sequence. For ordinary Kronecker sequences, such a relation is known in only one case, that of the (nφ)-sequence. The original motivation for our study of Hankel matrices was to recast the search for counterexamples to variants of Littlewood's conjecture over fields of Laurent series as the search for specific Kronecker-type sequences. Interestingly, the method developed for the study of Kronecker-type sequences also allows a new perspective on the celebrated Sobol' construction. Our approach yields a short and elegant proof of its excellent distribution properties.
ER  -

Abstract

Related

Cite