On a bounded remainder set for a digital Kronecker sequence

Levin, Mordechay B.

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 163-187.

Soit ${\mathbf{x}}_0,{\mathbf{x}}_1,...$ une suite de points dans ${[0,1)}^s$. Un sous-ensemble $S$ de ${[0,1)}^s$ est appelé un ensemble à restes bornés s’il existe un nombre réel $C$ tel que, pour tout entier positif $N$,

$$|\mathrm{card}\{n<N:{\mathbf{x}}_n\in S\}-\mathrm{mes}\left(S\right)N|<C.$$

Soient ${\left({\mathbf{x}}_n\right)}_{n\ge 0}$ une suite de Kronecker de dimension $s$ en base $b\ge 2$ et $\gamma =({\gamma }_1,...,{\gamma }_s)$, où, pour $i=1,...,s$, le développement en base $b$ de ${\gamma }_i\in [0,1)$, ${\gamma }_i={\gamma }_{i,1}{b}^{-1}+{\gamma }_{i,2}{b}^{-2}+\cdots$, vérifie ${\gamma }_{i,j}\ne b-1$ pour une infinité de $j$. Dans cet article, nous prouvons que $[0,{\gamma }_1)\times \cdots \times [0,{\gamma }_s)$ est un ensemble à restes bornés relativement à la suite ${\left({\mathbf{x}}_n\right)}_{n\ge 0}$ si et seulement si

$$\underset{1\le i\le s}{max}sup\{j\ge 1:{\gamma }_{i,j}\ne 0\}<\infty .$$

Nous obtenons ce résultat en conséquence d’un énoncé plus général donné dans la Proposition.

Let ${\mathbf{x}}_0,{\mathbf{x}}_1,...$ be a sequence of points in ${[0,1)}^s$. A subset $S$ of ${[0,1)}^s$ is called a bounded remainder set if there exists a real number $C$ such that, for every positive integer $N$,

$$|\mathrm{card}\{n<N:{\mathbf{x}}_n\in S\}-\mathrm{meas}\left(S\right)N|<C.$$

Let ${\left({\mathbf{x}}_n\right)}_{n\ge 0}$ be an $s$-dimensional digital Kronecker sequence in base $b\ge 2$, $\gamma =({\gamma }_1,...,{\gamma }_s)$, ${\gamma }_i\in [0,1)$ with base-$b$ expansion

${\gamma }_i={\gamma }_{i,1}{b}^{-1}+{\gamma }_{i,2}{b}^{-2}+\cdots$ for infinitely many ${\gamma }_{i,j}\ne b-1$, $i=1,...,s$. In this paper, we prove that $[0,{\gamma }_1)\times \cdots \times [0,{\gamma }_s)$ is a bounded remainder set with respect to the sequence ${\left({\mathbf{x}}_n\right)}_{n\ge 0}$ if and only if

$$\underset{1\le i\le s}{max}sup\{j\ge 1:{\gamma }_{i,j}\ne 0\}<\infty .$$

We get this result as a consequence of a more general statement given in the Proposition.

Classification : 11K38
Mots clés : bounded remainder set, digital Kronecker sequence

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