Variants of the Littlewood conjecture, their connection to uniformly distributed sequences, and the exact order of the discrepancy of van der Corput–Kronecker-type sequences
Roswitha Hofer1
1Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 489-514

The aims of this paper are twofold. First, it discusses the Littlewood conjecture and its variants with respect to uniformly distributed sequences. The second aim is to determine the exact order of the discrepancy of the van der Corput–Kronecker-type sequences which are based on recent counterexamples to the $X$-adic Littlewood conjecture over finite fields. Our result on the exact order of the discrepancy supports the well-established conjecture in the theory of uniform distribution, which states that $D_N\le c \frac{\log ^s N}{N}$, with $c>0$ for all $N>1$ is the best possible upper bound for the discrepancy $D_N$ of a sequence in $[0,1)^s$.

Les objectifs de cet article sont doubles. Premièrement, il traite de la conjecture de Littlewood et de ses variantes en relation avec les suites équidistribuées. Le second objectif est de déterminer l’ordre exact de la discrépance des suites de type van der Corput–Kronecker qui sont basées sur des contre-exemples récents à la conjecture de Littlewood $X$-adique sur les corps finis. Notre résultat sur l’ordre exact de la discrépance étaye la conjecture bien établie dans la théorie de l’équidistribution, selon laquelle $D_N\le c \frac{\log ^s N}{N}$, avec $c>0$ pour tout $N>1$, constitue la meilleure borne supérieure possible pour la discrépance $D_N$ d’une suite dans $[0,1)^s$.

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DOI : 10.5802/jtnb.1369
Classification : 11K31, 11K38, 11J13
Keywords: Variants of the Littlewood conjecture, van der Corput–Kronecker-type sequences, discrepancy, lower bounds

Roswitha Hofer  1

1 Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Roswitha Hofer. Variants of the Littlewood conjecture, their connection to uniformly distributed sequences, and the exact order of the discrepancy of van der Corput–Kronecker-type sequences. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 489-514. doi: 10.5802/jtnb.1369
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