She wont give me back my security deposit ?

Hi,I'm currently experiencing some problems in my new apartment. Before I left to summer brake in April I singed a written letter made by my future roommate. The letter said that I would be moving in August and would be paying the security deposit and August rent.I recently arrived to my apartment August 14 and experienced some trouble with the cockroaches and my roommate. She was violating my privacy by going inside my room to turn of the air conditioner when I was not home even though electricity was included in the written document,as well as graving my cooking utensils with out my permission and on top of that there were cockroaches crawling everywhere. After a few days, August 20, I told her that I will be moving out at the end of August and will be needing my security deposit back due to the cockroaches.She then continued to disagree because I had not given her a months notice.I would have but I did not arrive to the apartment until the 14 of August.Can you guys be nice enough to give me some law advice with links.One more thing I live in Arizona if that might help.One more thing the contract was a peace of paper that we signed it was not a lease and it said that it would be term by term basis.