This is the answer here. Even if you think it wasn’t a big deal apologize and try to build a bridge. It‘ll go a long way. I wouldn’t write to everyone, just your neighbor who complained directly to you.

I’m assuming everything you said in the post is true but make sure your kids are not playing in the halls and leaving toys out. If this person can’t be reasoned with and keeps filing complaints that are fabricated ask for proof (pictures of toys, recording of excessive noise). Obviously in that situation you had better be in the right.

Talk to your kids about keeping their voices down in the halls and have them do their best. There will be slip ups but as long as you’re just coming and going it is what it is.