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In this paper we extend a result of Levin, who proved a lower bound on the star discrepancy of generalized Halton sequences in positive integer bases, to rational bases. [ABSTRACT FROM AUTHOR]

Mordechay B. Levin (1999) has constructed a number λ which is normal in base 2, and such that the sequence ({ 2 n λ }) n = 0 , 1 , 2 , ... has very small discrepancy N ⋅ D N = O ((log ⁡ N) 2). This construction technique was generalized by Becher and Carton (2019), who generated normal numbers via nested perfect necklaces, for which the same upper discrepancy estimate holds. In this paper we derive an upper discrepancy bound for so-called semi-perfect nested necklaces and show that for Levin's normal number in arbitrary prime base p this upper bound for the discrepancy is best possible. This result generalizes a previous result by the authors (2022) in base 2. Our result for Levin's normal number in any prime base might support the guess that O ((log ⁡ N) 2) is the best order in N that can be achieved by a normal number, while generalizing the class of known normal numbers by introducing semi-perfect necklaces on the other hand might help for the search of normal numbers that satisfy smaller discrepancy bounds. [ABSTRACT FROM AUTHOR]

In this paper we investigate the distribution properties of hybrid sequences which are made by combining Halton sequences in the ring of polynomials and digital Kronecker sequences. We give a full criterion for the uniform distribution and prove results on the discrepancy of such hybrid sequences. [ABSTRACT FROM AUTHOR]

In this paper we investigate a special sequence related to the Halton sequence, namely the Halton sequence indexed by ⌊ n β ⌋ with β ∈ R , and prove a metric almost low-discrepancy result. [ABSTRACT FROM AUTHOR]

We discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research. [ABSTRACT FROM AUTHOR]

We study the distribution properties of sequences which are a generalization of the well-known van der Corput–Halton sequences on one hand, and digital (T,s)-sequences on the other. In this paper, we give precise results concerning the distribution properties of such sequences in the s-dimensional unit cube. Moreover, we consider subsequences of the above-mentioned sequences and study their distribution properties. Additionally, we give discrepancy estimates for some special cases, including subsequences of van der Corput and van der Corput–Halton sequences. [ABSTRACT FROM AUTHOR]