On a bounded remainder set for a digital Kronecker sequence

Mordechay B. Levin

doi:10.5802/jtnb.1197

Mordechay B. Levin ¹

¹ Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel

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

Résumé
Abstract

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} \in S} - mes (S) N | < C .

Soient ${(x_{n})}_{n \geq 0}$ une suite de Kronecker de dimension $s$ en base $b \geq 2$ et $γ = (γ_{1}, ..., γ_{s})$ , où, pour $i = 1, ..., s$ , le développement en base $b$ de $γ_{i} \in [0, 1)$ , $γ_{i} = γ_{i, 1} b^{- 1} + γ_{i, 2} b^{- 2} + \dots$ , vérifie $γ_{i, j} \neq b - 1$ pour une infinité de $j$ . Dans cet article, nous prouvons que $[0, γ_{1}) \times \dots \times [0, γ_{s})$ est un ensemble à restes bornés relativement à la suite ${(x_{n})}_{n \geq 0}$ si et seulement si

max_{1 \leq i \leq s} sup {j \geq 1 : γ_{i, j} \neq 0} < \infty .

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

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} \in S} - meas (S) N | < C .

Let ${(x_{n})}_{n \geq 0}$ be an $s$ -dimensional digital Kronecker sequence in base $b \geq 2$ , $γ = (γ_{1}, ..., γ_{s})$ , $γ_{i} \in [0, 1)$ with base- $b$ expansion

$γ_{i} = γ_{i, 1} b^{- 1} + γ_{i, 2} b^{- 2} + \dots$ for infinitely many $γ_{i, j} \neq b - 1$ , $i = 1, ..., s$ . In this paper, we prove that $[0, γ_{1}) \times \dots \times [0, γ_{s})$ is a bounded remainder set with respect to the sequence ${(x_{n})}_{n \geq 0}$ if and only if

max_{1 \leq i \leq s} sup {j \geq 1 : γ_{i, j} \neq 0} < \infty .

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

Reçu le : 2020-09-10
Révisé le : 2021-11-10
Accepté le : 2021-12-01
Publié le : 2022-07-07

DOI : 10.5802/jtnb.1197

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

Affiliations des auteurs :

Mordechay B. Levin ¹

¹ Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
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Mordechay B. Levin. On a bounded remainder set for a digital Kronecker sequence. Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 163-187. doi : 10.5802/jtnb.1197. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1197/

Bibliographie
Cité par

[1] Boris Adamczewski; Yann Bugeaud On the Littlewood conjecture in fields of power series, Probability and number theory - Kanazawa 2005 (Advanced Studies in Pure Mathematics), Volume 8, Mathematical Society of Japan, 2005, pp. 1-20 | Zbl

[2] Faustin Adiceam; Erez Nesharim; Fred Lunnon On the $t$ -adic Littlewood Conjecture, Duke Math. J., Volume 170 (2021) no. 10, pp. 2371-2419 | MR | Zbl

[3] József Beck Probabilistic diophantine approximation. I: Kronecker sequences, Ann. Math., Volume 140 (1994) no. 1, pp. 109-160 | DOI | MR

[4] József Beck; William W. L. Chen Irregularities of Distribution,, Cambridge Tracts in Mathematics, 89, Cambridge University Press, 1987 | DOI

[5] Dmitriy Bilyk; Michael T. Lacey; Armen Vagharshakyan On the small ball inequality in all dimensions, J. Funct. Anal., Volume 254 (2008) no. 9, pp. 2470-2502 | DOI | MR | Zbl

[6] Josef Dick; Friedrich Pillichshammer Digital Nets and Sequences, Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, 2010 | DOI

[7] Sigrid Grepstad; Nir Lev Sets of bounded discrepancy for multi-dimensional irrational rotation, Geom. Funct. Anal., Volume 25 (2015) no. 1, pp. 87-133 | DOI | MR | Zbl

[8] Peter Hellekalek Regularities in the distribution of special sequences, J. Number Theory, Volume 18 (1984) no. 1, pp. 41-55 | DOI | MR | Zbl

[9] Peter Hellekalek General discrepancy estimates: the Walsh function system, Acta Arith., Volume 67 (1994) no. 3, pp. 209-218 | DOI | MR | Zbl

[10] Roswitha Hofer Kronecker-Halton sequences in $𝔽_{p} ((X^{- 1}))$ , Finite Fields Appl., Volume 50 (2018), pp. 154-177 | DOI | MR | Zbl

[11] Gerhard Larcher On the distribution of an analog to classical Kronecker-sequences, J. Number Theory, Volume 52 (1995) no. 2, pp. 198-215 | DOI | MR | Zbl

[12] Gerhard Larcher Digital Point Sets: Analysis and Applications, Random and quasi-random point sets (Lecture Notes in Statistics), Volume 138, Springer, 1998, pp. 167-222 | DOI | MR | Zbl

[13] Gerhard Larcher Probabilistic Diophantine approximation and the distribution of Halton–Kronecker sequences, J. Complexity, Volume 29 (2013) no. 6, pp. 397-423 | DOI | MR | Zbl

[14] Gerhard Larcher; Harald Niederreiter Kronecker-type sequences and nonarchimedean Diophantine approximations, Acta Arith., Volume 63 (1993) no. 4, pp. 379-396 | DOI | Zbl

[15] Gerhard Larcher; Harald Niederreiter Generalized $(t, s)$ -sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series, Trans. Am. Math. Soc., Volume 347 (1995) no. 6, pp. 2051-2073 | MR | Zbl

[16] Gerhard Larcher; Friedrich Pillichshammer Metrical lower bounds on the discrepancy of digital Kronecker-sequences, J. Number Theory, Volume 135 (2014), pp. 262-283 | DOI | MR | Zbl

[17] Mordechay B. Levin On the lower bound in the lattice point remainder problem for a parallelepiped, Discrete Comput. Geom., Volume 54 (2015) no. 4, pp. 826-870 | DOI | MR | Zbl

[18] Mordechay B. Levin On the lower bound of the discrepancy of Halton’s sequences. I, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 5, pp. 445-448 | DOI | MR

[19] Mordechay B. Levin On the lower bound of the discrepancy of $(t, s)$ -sequences. I., C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 6, pp. 562-565 | DOI | MR | Zbl

[20] Mordechay B. Levin On a bounded remainder set for $(t, s)$ -sequences. I, Chebyshevskiĭ Sb., Volume 20 (2019) no. 1, pp. 222-246 | DOI

[21] Rudolf Lidl; Harald Niederreiter Introduction to Finite Fields and their Applications, Cambridge University Press, 1994 | DOI

[22] Harald Niederreiter Random Number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, 63, Society for Industrial and Applied Mathematics, 1992 | DOI

[23] Maxim M. Skriganov Construction of uniform distributions in terms of geometry of numbers, Algebra Anal., Volume 6 (1994) no. 3, pp. 200-230 | MR | Zbl

[24] Donald C. Spencer The lattice points of tetrahedra, J. Math. Phys., Volume 21 (1942), pp. 189-197 | DOI | MR | Zbl

Cité par Sources :