What is the rate of decay of $\min\{k\xi-\lfloor k\xi\rfloor|k\in\{1,\dots,n\}\}$, for irrational $\xi$?

Question

I wish to establish bounds on the sequence of infima of $\{n\xi\}_{n\in\Bbb N}$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part function and $\xi$ is irrational.

I can prove that $\liminf_{n\to\infty}\{n\xi\}=0$ by showing that for any $k\in\Bbb N$, there exists $n\in\Bbb N$ such that $\{n\xi\}<1/k$. Consider $\{\{\xi\},\{2\xi\},\dots,\{(k+2)\xi\}\}$ for some $k\in\Bbb N$ (this set has no duplicates and misses rationals because $\xi$ is irrational). If we put these $k+1$ pigeons into the $k$ holes $[i/k,(i+1)/k)$ (for $0\le i<k$), we know that at least two members $\{i\xi\},\{j\xi\}$ satisfy $0<\{i\xi\}-\{j\xi\}<1/k$, so that $\{(i-j)\xi\}\in(0,1/k)$. If $i>j$, we are done (with $n=i-j$). Otherwise, let $m=\lfloor1/\{(i-j)\xi\}\rfloor$. Then $$\{m(i-j)\xi\}=m\{(i-j)\xi\}>1-\frac1{\{(i-j)\xi\}}>1-\frac1k,$$

so $\{m(j-i)\xi\}=1-\{m(i-j)\xi\}<1/k$ and we let $n=m(j-i)$.

This method of proof is good for showing that $\{n\xi\}_{n\in\Bbb N}$ is dense in $[0,1]$, but it provides no bound at all on the rate of decay (since $m$ can be arbitrarily large). What bounds are known on this quantity (independent of $\xi$)?

Answer 1

There is a theorem on asymmetric Diophantine approximations:

Serge (1945)
For any $\xi \in \mathbb{R}\setminus\mathbb{Q}$ and any $\tau \ge 0 $, there exists infinitely many $\frac{p}{q} \in \mathbb{Q}$ such that $$-\frac{1}{\sqrt{1+4\tau} q^2} < \xi - \frac{p}{q} < \frac{\tau}{\sqrt{1+4\tau}q^2}$$ Furthermore, the statement holds when $\displaystyle \xi - \frac{p}{q}$ is replaced by $\displaystyle \frac{p}{q} - \xi$.

Taking $\tau = 0$, we find for any irrational $\xi$, there are infinitely many rational $\displaystyle \frac{p}{q}$ such that

$$0 < \xi - \frac{p}{q} < \frac{1}{q^2} \iff \{q\xi\} = \xi - \lfloor q \xi\rfloor < \frac{1}{q} $$

This implies $$\liminf_{n\to\infty} ( n \{ n \xi \} ) \le 1.$$

On the other hand, Eggan and Niven (1961) has shown for any $\gamma > 1$, there are $\eta \in \mathbb{R}\setminus\mathbb{Q}$ such that the inequalities $$0 < \eta - \frac{a}{b} < \frac{1}{\gamma b^2}, \quad \frac{a}{b} \in \mathbb{Q}$$ has no rational solution. For such $\eta$, we have: $$\liminf_{n\to\infty} ( n \{ n \eta \} ) \ge \gamma$$ Combine these two known facts, we have

$$\sup\Big\{\;\liminf_{n\to\infty} ( n \{ n \xi \} ) : \xi \in \mathbb{R}\setminus\mathbb{Q}\;\Big\} = 1 $$ So in certain sense, the "best" bound for infima of $\{ n \xi \}$ independent of $\xi$ is really $\displaystyle \frac{1}{n}$.

For more info and references, see Section 1.3-1.4 of Ivan's Niven's book

Diophantine Approximations (Dover Books on Mathematics), ISBN 978-0486462677