Downsampling and Gaussian Filtering in the Context of Scale Space Pyramids
You're correct, it has to do with the Cut Off frequency of the Gaussian Blur Filter in its Frequency Domain.  

In order to see it, just apply a DFT (Using MATLAB it can be achieved by fft / fft2) and look on the absolute value.
Look for the -3dB point and you'll see.

There is also an intuitive explanation on the original article which say that blurring using Gaussian Kernel with $ \sigma $ means removing any detail which its size is smaller than $ \sqrt{ \sigma } $.