I'm having the same problem, and am starting to suspect that we've been tricked into sending small change to an address not under our own control. I can't find evidence on my own phone that it's actually downloading the blockchain (headers or full). There is no block count, only a spinning icon; so there is no way to judge it's progress. The only option in the settings section is 'refresh'. What is it refreshing? I can't examine the code, so until I see evidence otherwise, or someone on this forum that I trust will inform me otherwise, I will consider this a rather ingenious method of 'phishing' fraud.
I am here to say, with certainty, that this is not a fraud. I got mine to work, and now I think I know what has been going wrong. The download is getting interupted without any kind of error or even a force close. To get it done, you need to plug your phone into the charger, turn on the app, select 'refresh' from the settings menu, and just set it down near your wifi router and go to bed. DO NOT hit the button to make the screen sleep. The app will keep the screen on, which is why you need it plugged in, and will take many hours to march through the blockchain. But by morning, the spinning icon should be gone and any coins that you sent to it should be visable. You can safely stop the app, and the next time you start the app, it will take only a short while to catch up.