Metrical lower bounds on the discrepancy of digital Kronecker-sequences

2013/02/21 by Gerhard Larcher, Larcher, Gerhard, Friedrich Pillichshammer +1
Computer Science · Mathematics · #11K06 #11K38 #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1302.5267

openalex publication_date 2013/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Digital Kronecker-sequences are a non-archimedean analog of classical Kronecker-sequences whose construction is based on Laurent series over a finite field. In this paper it is shown that for almost all digital Kronecker-sequences the star discrepancy satisfies D_N^∗ ≥ c(q,s) (log N)^s log log N for infinitely many N ∈ \NN, where c(q,s)>0 only depends on the dimension s and on the order q of the underlying finite field, but not on N. This result shows that a corresponding metrical upper bound due to Larcher is up to some log log N term best possible.

Citations

• Point sets and sequences with small discrepancy

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