Contrary to popular belief, you have not achieved much when you get a user to download your app. Usage is the only thing that counts, and yes, download and install are preconditions, but not much more than that. So to many app providers, push notifications are vital to ensure an ongoing app usage. Considering the importance of the field, there are suprisingly few studies. Maybe that's because the results are all too obvious, like in this case:
Users only want to receive push notifications that are relevant, time-sensitive and valuable.