Let be a sequence of points in . A subset of is called a bounded remainder set if there exists a real number such that, for every positive integer ,
Let be an -dimensional digital Kronecker sequence in base , , with base- expansion
for infinitely many , . In this paper, we prove that is a bounded remainder set with respect to the sequence if and only if
We get this result as a consequence of a more general statement given in the Proposition.
Soit une suite de points dans . Un sous-ensemble de est appelé un ensemble à restes bornés s’il existe un nombre réel tel que, pour tout entier positif ,
Soient une suite de Kronecker de dimension en base et , où, pour , le développement en base de , , vérifie pour une infinité de . Dans cet article, nous prouvons que est un ensemble à restes bornés relativement à la suite si et seulement si
Nous obtenons ce résultat en conséquence d’un énoncé plus général donné dans la Proposition.
Revised:
Accepted:
Published online:
Mots-clés : bounded remainder set, digital Kronecker sequence
Mordechay B. Levin 1

@article{JTNB_2022__34_1_163_0, author = {Mordechay B. Levin}, title = {On a bounded remainder set for a digital {Kronecker} sequence}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {163--187}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {34}, number = {1}, year = {2022}, doi = {10.5802/jtnb.1197}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1197/} }
TY - JOUR AU - Mordechay B. Levin TI - On a bounded remainder set for a digital Kronecker sequence JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 163 EP - 187 VL - 34 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1197/ DO - 10.5802/jtnb.1197 LA - en ID - JTNB_2022__34_1_163_0 ER -
%0 Journal Article %A Mordechay B. Levin %T On a bounded remainder set for a digital Kronecker sequence %J Journal de théorie des nombres de Bordeaux %D 2022 %P 163-187 %V 34 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1197/ %R 10.5802/jtnb.1197 %G en %F JTNB_2022__34_1_163_0
Mordechay B. Levin. On a bounded remainder set for a digital Kronecker sequence. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 163-187. doi : 10.5802/jtnb.1197. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1197/
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