Can you bring an empty water bottle with you on the plane?

Yes, do this all the time.

Probably not necessary, but I take the lid off and put it in another pocket to ensure it's seen as an empty bottle.

UK airports generally have a bar airside and I've never had a problem asking them to refill the bottle.

In the United States, the website of the TSA specifically states:

You may take empty drink containers through the checkpoint and fill them afterward.

Unfortunately, I didn't see a way to link directly to the page containing that text, but if you go to the link above and then type "empty water bottle" into the "Can I bring my..." search box, it will display the above text.

Most (all?) U.S. airports have water fountains airside where you can fill up your water bottle after going through security. Hong Kong and Incheon both had them, as well. Some airports even have water fountains specifically designed for filling up water bottles (where you don't have to tilt the bottle over to get water in it.) I've seen these both in Nashville and in Phoenix.

The only weird thing I've ever encountered with this was one time on a flight from Hong Kong to the U.S. There were actually security personnel inside the jetway in Hong Kong that were going through everyone's bags and confiscating non-empty water bottles, including the ones that people had purchased inside the concourse. For more information on this see the questions:

Why were water bottles confiscated in a boarding gate jetway in Hong Kong?
When are U.S.-bound flights required to confiscate liquids at the gate?

I have been using cheap(ish) drinking bottles as well as re-used 'one time' water bottles for going through security in Europe.
Mostly I have the bottle on the outside of my luggage or place it in the bin next to my laptop or shoes.

Several times I have been asked whether the bottle is empty, on the answer: "as empty as I can get it by drinking" it was allowed to go through.

But no single airport security site I have seen so far guarantees the passage of empty water bottles and I would not take my expensive water bottle in hand luggage.
Most of the time it will not be a problem at all but when you run into the one officer or one airport that does not allow it through, losing a cheap bottle will hurt less than losing an expensive or loved bottle.

By the way, I find it no hardship to drink half a liter of water within a few minutes and have done so, like when I met a security check where I did not expect one. Waiting outside the line, although at that point in time there was very little line to wait next to.
The security officer on my side of the system offered to empty the bottle into their bin and return it to me, I declined as I was in need of a drink by then.

It depends on where you are in the world. In Britain there are rarely places to fill bottles up, yet they are allowed through. In a way, I think its a moneymaking scam, if you have liquids in the bottles at UK security the bottle goes, in Sweden the liquid goes. In Britain they probably sell the bottles

Though security might let you get away with this, regulations clearly state (UK, but the same applies elsewhere) that vessels that are able to contain more than 100ml, even when they contain less (or nothing at all) are not allowed through security.

Case in point: have you ever tried to take a half empty tube of toothpaste through security?