Can please someone help me with the following problem.


Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational.


Now I need to solve the inequality
$nx \; \mathrm{mod} \; 1 < v$
with respect to $n$, for some given small $v$.


According to Equidistribution Theorem, this sequence is uniformly distributed on (0,1). And thus we definitely know that there is an infinite number of those $n$'s. But what can we say about these solutions themselves? In particular, I need to know if the distance between two consequtive solutions is limited or not. 


It seems to be true to me, but maybe I'm missing something... and can't figure out a hard proof.



Equidistribution Theorem: distance between solutions
Here is a quick solution to your problem. Fix a positive integer $q$ such that $\{ qx \} <v $, and then fix a positive integer $s$ such that $1<s\{qx\}$. Now assume that $n$ is any integer satisfying $\{nx\}<v$. Let $r$ be the smallest positive integer such that $1<\{nx\}+r\{qx\}$. Clearly, $r\leq s$. In addition, $\{nx\}+r\{qx\}<1+\{qx\}<1+v$, hence in fact $\{(n+rq)x\}<v$. We showed that the gaps between the solutions of $\{nx\}<v$ is at most $sq$.


