Dear {Recipient},
As your neighbor, I would like to apologize for {problem}. It has come to my attention that this problem has really been bothering you, and I simply had no idea. I have taken steps to prevent {problem} in the future, and I hope you will feel free to contact me if there's ever anything like this bothering you ever again. You can reach me at{phone/email}, if you didn't know.
I like to think that we are a neighborhood, and that we can work these things out in a friendly fashion. So, please accept my apology for {problem} and rest assured that had I known about this issue sooner, I would have stopped immediately.
Your neighbor,
{Sender}
