How to Use BTsync's file sharing by Bluetooth? I established the bluetooth connection between the devices. 
However, the speed download/upload still stays low (&lt; 1 MB/s) between the devices while it should be much higher with Bluetooth 4.0 in both devices: Macbook Air Mid-2013 (adapter) and Samsung Galaxy S4. 
BTsync should use local network to share data between the devices in the sharing. 

I contacted the BTsync support about the issue 


  Unfortunately we don't support bluetooth
  so Sync establishes connection through LAN or internet, if there are
  firewalls and NATs it can be established through relay server. But
  Sync uses the best possible connection for synchronization. We'll
  don't plan supporting bluetooth in nearest future but we'll think
  about it.


So it is completely technically possible to support file sharing in the local network by Bluetooth. 



How can you enable BTsync file sharing by bluetooth in OS X with Android?
This feature request is now active in their development. 
However, it can take some years to get it because of limited resources in their development. I will let you know when I get more pieces of information about the issue. There are some tickets which has to be solved before such a feature can be implemented.