There are a number of sites such as
this one, which claim that filtering cheap vodka through the Brita water system will render the product as good as top shelf, such as Grey Goose. Most sites I've seen are from college students and where the fun begins and objective comparison ends is a little hard to tell. However, I believe they're onto something and it's worth the effort to try.
I don't drink all that much vodka but I'm still considering trying it. One thing never mentioned is preparing the filter. It requires flushing with clean water when starting its work cycle, thus the water-logged filter would need to be displaced with vodka before capturing any filtered product.
I would imagine the filter could be stored in the freezer between usage.
If I try it I'll post the results.