Sums of reciprocals of fractional parts and multiplicative Diophantine approximation

TY - JOUR

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

AU - Beresnevich, Victor

AU - Haynes, Alan

AU - Velani, Sanju

N1 - © 2020, American Mathematical Society 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 - 2020/2/24

Y1 - 2020/2/24

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.

U2 - 10.1090/memo/1276

DO - 10.1090/memo/1276

M3 - Article

SN - 0065-9266

VL - 263

SP - 11

EP - 24

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

IS - 1276

ER -