Is your note 3 battery life very bad ? Mine never use much the whole day from full charge now only 20% left
Some possibilities:
1. A rogue app wakelocking your phone. Some live wallpapers can do this too. I once installed a beautiful sakura live wallpaper that killed my S3's battery in record time because it was constantly sending data out for some reason. Uninstalled asap.
2. Poor network connection causing battery drain while trying to find a stable connection.
3. If you're connected to a wifi network, something else on the network might be constantly pinging your phone, wakelocking it.
4. "Location services -> Use wireless networks" option is on. This thing is known to wakelock your phone, and gets worse if you have more apps that keep asking for your phone's location. I've seen it keep my old S3 awake for 2.5 hours before.
Unfortunately, to recognize which problem it is, the easiest way is to install BetterBatteryStats (which is not free iirc) and post the logs up here (or at XDA), then we can advise you on the app(s) causing the battery drain.
My own N3 went from full charge to about 65% after about 10 hours with moderate s pen usage (s pen usage increases battery consumption) and it's constantly paired with the Galaxy Gear.