sum of fractional parts (nx_i),x_i are irrational Let $x_1,x_2,...,x_k$ be irrational number,is it always true that:
$\liminf_{n\rightarrow\infty} \sum_{i=1}^k (nx_i)=0$ (where $(x)$ denotes the fractional part of $x$)
If not,what are the necessary and sufficient conditions that {$x_i$} must satisfy so that $\liminf_{n\rightarrow\infty} \sum_{i=1}^k (nx_i)=0$ 
 A: Here is a counterexample: $x_1 = \sqrt2, x_2 = 1-\sqrt2$.
For $n>0$, $(nx_1) + (n x_2) =1$ so the limit infimum is $1$, not $0$.
I think the other answers assumed that you meant $x-[x]$, where $[x]$ is the nearest integer to $x$, instead of $(x)$. 
If there is no rational dependency, then the multiples are dense in the torus $(\mathbb{R}/\mathbb{Z})^k$, so the limit infimum is $0$. If there is some rational dependency, there will always be small multiples (close to the origin in $(\mathbb{R}/\mathbb{Z})^k$), but these might not be positive.
Consider the closure of multiples of $\langle x_1,...,x_k \rangle$ in the torus $(\mathbb{R}/\mathbb{Z})^k$. This is a closed subgroup, hence a finite union of parallel flat subtori. For $(x)$, the condition you want is that the tangent space to this subgroup at the origin intersects the positive orthant.
A: If you take any vector in Euclidean space, and look at its integer multiples modulo the integer lattice, we know what happens qualitatively. They are dense in a certain subtorus of the obvious torus. That's a version of Kronecker's theorem. Your question is about being near the origin infinitely often. So I say it's true.
A: Yes. This is the subject of "simultaneous diophantine approximation", googling which will produce many results, but you can take a look at Doug Hensley's notes:
http://www.math.tamu.edu/~Doug.Hensley/SimultaneousDiophantine.pdf
