On the fractional parts of certain sequences of $\xi \alpha^{n}$
Abstract
Assume that $\alpha>1$ is an algebraic number and $\xi\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(\xi \alpha^{n})$. Under various Diophantine conditions on $\xi$ and $\alpha$, we obtain lower bounds on the number $n$ with $1\leq n\leq N $ for which the fractional part of the sequence $(\xi \alpha^{n})_{n\geq1}$ fall into a prescribed region $I\subset [0,1]$, extending several results in the literature. As an application, we show that the Fourier decay rate of some self-similar measures is logarithmic, generalizing a result of Varjú and Yu.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.02972
- arXiv:
- arXiv:2408.02972
- Bibcode:
- 2024arXiv240802972G
- Keywords:
-
- Mathematics - Number Theory;
- Mathematics - Classical Analysis and ODEs;
- Primary: 11J71;
- 28A80. Secondary: 11K16;
- 37A45;
- 42A38
- E-Print:
- 12 pages