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.

Let x 0 ,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,

| card {n<N:x n S}- meas (S)N|<C.

Let (x n ) n0 be an s-dimensional digital Kronecker sequence in base b2, γ=(γ 1 ,...,γ s ), γ i [0,1) with base-b expansion

γ i =γ i,1 b -1 +γ i,2 b -2 + for infinitely many γ i,j b-1, i=1,...,s. In this paper, we prove that [0,γ 1 )××[0,γ s ) is a bounded remainder set with respect to the sequence (x n ) n0 if and only if

max 1is sup{j1:γ i,j 0}<.

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

Soit x 0 ,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,

| card {n<N:x n S}- mes (S)N|<C.

Soient (x n ) n0 une suite de Kronecker de dimension s en base b2 et γ=(γ 1 ,...,γ s ), où, pour i=1,...,s, le développement en base b de γ i [0,1), γ i =γ i,1 b -1 +γ i,2 b -2 +, vérifie γ i,j b-1 pour une infinité de j. Dans cet article, nous prouvons que [0,γ 1 )××[0,γ s ) est un ensemble à restes bornés relativement à la suite (x n ) n0 si et seulement si

max 1is sup{j1:γ i,j 0}<.

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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1197
Classification: 11K38
Keywords: bounded remainder set, digital Kronecker sequence
Mordechay B. Levin 1

1 Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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