Hankel Matrix Decompositions and Digital Sequences over Function Fields

Source 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 $\mathcal{L}\mathcal{B}\mathcal{U}$ decomposition, using recursive algorithms based on the continued-fraction coefficients of the corresponding Laurent series. The upper-triangular factor $\mathcal{U}$ has several remarkable properties. In particular, it is constructed columnwise from the so-called Ostrowski-type representations of powers of $X$. Applying $\mathcal{U}$ 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 $(n\varphi)$-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.