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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 $\liminf_{n\rightarrow\infty} \sum_{i=1}^k (nx_i)=0$

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Oh sorry,I have made a mistake in the question.$(x)$ should be equal to $|x-[x]|$,where $[x]$ is the nearest integer to $x$. –  Ben May 21 '12 at 1:19
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3 Answers

up vote 4 down vote accepted

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.

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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.

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That is not exactly correct, since this only shows you are near a lattice point, but not necessarily near the origin. –  Igor Rivin May 19 '12 at 13:52
    
Yes, there's the good point I missed that where the numbers are rationally dependent there is the geometry explained in another answer. –  Charles Matthews May 20 '12 at 7:53
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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

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