# How should I understand the link between $x$ and $| x \rangle$?

In the paper "Quantum Expectation-Maximization for Gaussian Mixture Models" I encountered the following proposition :

Consider two vectors x,y and $$\theta$$ the angle between x and y, $$\theta < \frac{\pi}{2}$$. Then $$||x - y|| \leq \epsilon$$ implies $$|| |x\rangle - |y\rangle || \leq \frac{\sqrt 2 \epsilon}{||x||}$$.

I'm not sure how to interpret the link between $$x$$ and $$| x \rangle$$ ? In the case x is an integer, for example $$x = 4$$ then I'd say that $$| x \rangle = | 100 \rangle$$. But what about $$x = 3.5$$ or $$x = (2.5, 2.5)$$ ? In those cases what $$| x \rangle$$ would be ?

Generally it's context-specific what the label means. For example, if $$x$$ and $$y$$ are integers, then if $$x\neq y$$ then $$|x\rangle$$ and $$|y\rangle$$ are orthogonal (since they are different binary strings).
I haven't read the paper you mentioned but in the "Preliminaries" section of 3.1, they describe what they mean by $$| x\rangle$$. Specifically: They assume $$x\in \mathbb{R}^d$$, and they define $$|x\rangle = \frac{1}{\Vert x\Vert}\sum_{j=1}^d x_j|j\rangle$$
So the $$|j\rangle$$ will be computational basis vectors, i.e. for $$j=1$$, $$|j\rangle = |0\dots 01\rangle$$, etc. The components of $$x$$ give the amplitudes of each of those basis vectors in some superposition.
With this definition of $$|x\rangle$$, the "Claim 3.4" should follow.