Sums of reciprocals of fractional parts and multiplicative Diophantine approximation

Research output: Contribution to journalArticle

Standard

Sums of reciprocals of fractional parts and multiplicative Diophantine approximation. / Beresnevich, Victor; Haynes, Alan; Velani, Sanju.

In: Memoirs of the American Mathematical Society, 02.04.2017.

Research output: Contribution to journalArticle

Harvard

Beresnevich, V, Haynes, A & Velani, S 2017, 'Sums of reciprocals of fractional parts and multiplicative Diophantine approximation', Memoirs of the American Mathematical Society.

APA

Beresnevich, V., Haynes, A., & Velani, S. (Accepted/In press). Sums of reciprocals of fractional parts and multiplicative Diophantine approximation. Memoirs of the American Mathematical Society.

Vancouver

Beresnevich V, Haynes A, Velani S. Sums of reciprocals of fractional parts and multiplicative Diophantine approximation. Memoirs of the American Mathematical Society. 2017 Apr 2.

Author

Beresnevich, Victor ; Haynes, Alan ; Velani, Sanju. / Sums of reciprocals of fractional parts and multiplicative Diophantine approximation. In: Memoirs of the American Mathematical Society. 2017.

@article{567130cf5f184052933402e4d99c70fa,
title = "Sums of reciprocals of fractional parts and multiplicative Diophantine approximation",
abstract = "There are two main interrelated goals of this paper. Firstly we investigate the sums $S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}$ and $R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|},$where $\alpha$ and $\gamma$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W.M.Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of $\{1\le n\le N:\|n\alpha-\gamma\|<\ve\} \, ,$ valid for all sufficiently large $N$ and all sufficiently small $\ve$.This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in $\R^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.",
author = "Victor Beresnevich and Alan Haynes and Sanju Velani",
note = "This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details",
year = "2017",
month = "4",
day = "2",
language = "English",
journal = "Memoirs of the American Mathematical Society",
issn = "0065-9266",
publisher = "American Mathematical Society",

}

TY - JOUR

T1 - Sums of reciprocals of fractional parts and multiplicative Diophantine approximation

AU - Beresnevich, Victor

AU - Haynes, Alan

AU - Velani, Sanju

N1 - This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

PY - 2017/4/2

Y1 - 2017/4/2

N2 - There are two main interrelated goals of this paper. Firstly we investigate the sums $S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}$ and $R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|},$where $\alpha$ and $\gamma$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W.M.Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of $\{1\le n\le N:\|n\alpha-\gamma\|<\ve\} \, ,$ valid for all sufficiently large $N$ and all sufficiently small $\ve$.This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in $\R^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.

AB - There are two main interrelated goals of this paper. Firstly we investigate the sums $S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}$ and $R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|},$where $\alpha$ and $\gamma$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W.M.Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of $\{1\le n\le N:\|n\alpha-\gamma\|<\ve\} \, ,$ valid for all sufficiently large $N$ and all sufficiently small $\ve$.This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in $\R^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.

M3 - Article

JO - Memoirs of the American Mathematical Society

T2 - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

ER -