I'm reading this paper (https://arxiv.org/abs/1602.02399) where the author talks about $\mathcal{M}$-resonant vectors that are $\tau$-Diophantine. Could you explain why the set of Diophantine vectors behaves for different values of $\tau$ the way it is claimed below to be?
Here is a screenshot of page 86 of the paper.
Any proof or hint to a proof for the simpler cases, for example when $m = 0$, is also appreciated.
Update: Let us consider the case $m = 0$. Yesterday, I was thinking that if we take $\omega^\perp$ to be a unit vector perpendicular to $\omega$ and if $q$ is a rational vector with $|q - \omega^\perp| < \delta$ with $\omega^\perp$, then
$$ |q.\omega| = |(q - \omega^\perp).\omega| < \delta|\omega|; $$
Therefore, if $\delta := \epsilon / \omega$, then $|q.\omega| < \epsilon$. Then we can multiply $q = \left(\dfrac{r_1}{s_1}, ..., \dfrac{r_n}{s_n}\right)$ by the least common multiple of $s_1, ..., s_n$. Let us call that number $s$ and assume the denominators are natural numbers without loss of generality. Then $z:= s\left(\dfrac{r_1}{s_1}, ..., \dfrac{r_n}{s_n}\right) = (z_1, ..., z_n)$ is an integer vector with $|z - s\omega^\perp| < s\delta$. This implies $|z.\omega| = |(z - s\omega^\perp).\omega| < s\delta.|\omega| = s\epsilon.$ We can have an estimate on the magnitude of $z$ as well. One can say $|z| = s|q|$, and $\delta > |q - \omega^\perp| \geq 1 - |q|$. Therefore, $|q| > 1 - \delta$ and $|z| > s - s\epsilon/|\omega|$.
Motivated from the above, let us now define for $\epsilon > 0$ the smallest positive integer $s_\epsilon$ so that $|s\omega^\perp| < \epsilon/|\omega|$ in the flat torus $\mathbb{T}^n := \dfrac{\mathbb{R}^n}{\mathbb{Z}^n}$. Then we can left the curve $\{t\omega^\perp: 0 \le t \le s\}$ to $\mathbb{R}^n$ and fidn an integer vector $k_\epsilon$ with the properties
- $|k_\epsilon| > s_\epsilon\left( 1- \dfrac{\epsilon}{|\omega|}\right)$
- $\epsilon > |k_\epsilon.\omega| \geq \dfrac{\gamma}{|s_\epsilon|^{\tau}\left|1 - \dfrac{\epsilon}{|\omega|}\right|^\tau}$
If one can estimate $s_\epsilon$ in terms of $\epsilon$, then I assume they should be able to find conditions on $\tau$. So, basically, the question is related to the return times of the time-1 map of the linear flow $\varphi^t(x) = x + t\omega^\perp$ on the flat torus to small neighbourhoods of the $0$ point. In the simple case where $n = 1$, the question is about rotations on the circle and one can give simple arguments using the pegionhole principle. But how about higher dimensions?