A blank Redirect URI in a mobile app is hanging the app

Thanks for the post. As you mentioned, Xamarin isn't officially supported for the official Dropbox API v2 .NET SDK, so I can't offer much help with that, but I'll pass this along as a feature request. I can't promise if or when that might be implemented though. 

And by "Mobile apps don't need to set the Redirect URI in app console and the same need not be supplied while signing in" I think you're referring to how the official Dropbox API mobile SDKs (i.e., the official Dropbox API v2 Java SDK for Android, and Swift and Objective-C SDKs for iOS) don't require the app/developer to specify a redirect URI. That's only because these official SDKs have been written specially to implement the app authorization flow for you, and automatically utilize the official Dropbox mobile app, if installed, or the official Dropbox web site, if not.

If you're not using one of those official mobile SDKs though, such as in this case where you're using the .NET SDK, that special pre-built authorization flow doesn't apply. So, I wouldn't expect supplying an empty string as the redirect URI to work. When using the standard OAuth flow like this, the redirect URI should be the actual URI where you want to send the user after they authorize the app, to complete the flow. You can find information on how the OAuth help in the .NET SDK is meant to be used in the documentation here.