After using the tape or paper in the door trick to confirm someine is entering, consult with your landlord and inquire if the locks where changed before you moved in. (it would be REAL spooky if it was a past tenant) Ask if the landlord entered your apartment.
You may find there is a reaonsable explaination.
if response is unstatisfactory, advise landlord you will be installing remote hidden cameras. Very often the thought there is veriifcation or people watching will end the transgression. Then head to Cosco and buy the wireless camera(s).
You may want to visit the X10 website and find out about all the 'spy' stuff that is available and will connect to your cell. Be careful, it is easy to become the 'spooky' person who has all the spying stuff.