Question 169
Does there exist an irrational number $x$ such that there are at most finitely many positive integers $n$ satisfying
$$
\{kx\} \geq \frac{1}{n+1}
$$
for every $k \in \{1, \dots, n\}$?
Note: Here, for a positive real number $y$, $\{y\}$ denotes the fractional part of $y$.
$$
\{kx\} \geq \frac{1}{n+1}
$$
for every $k \in \{1, \dots, n\}$?
Note: Here, for a positive real number $y$, $\{y\}$ denotes the fractional part of $y$.