urgent violation of lease

hello,
i recieved a letter from my managing agent's attorney in ny that i am in default of my lease due to not having proper carpeting installed in 80% of my apartment. I have been living here for a year and half and signed a new 2 year lease in february. a week later a new nieghbor moved in downstairs and started complaing about the noise that my two children 1 and 4 years old are making. the managemnt sent us a letter to put extra carpeting and we called them and explained that since my son has asthma we have to keep the carpeting to a minumum. Also, I need to point out that the nieghbor below me is single in his 40's and we have 2 small children that move around (being kids). the person at the management sympathized with me and that was the end until today. I came back from vacation and was given a notice by a lawyer that i was in violation of my lease and my lease will be terminated by monday April 20th if my apartment will not be in compliance.
furthermore i would like to note that i have a smal note from my sons doctor stating that thick carpeting would aggravate his situation.
I would also like to note that there is a clause in our lease that says a tenant has the right to "quiet enjoyment". we have kids and this neighbor wants to go to war and have us evicted. even with the carpet installe there will still be noise because of very little space between the floors and wood floors. we rent from a very large building and the letter was from a law firm respresenting the management specializing in landlord tenant relations. I would prefer not going to court.
would one letter from the doctor be enough or must i get the carpet installed. would it be better to be evicted and find a new place to live or try to do as they ask.
Please, if there is anyone who may offer advice i would appreciate it.

Thank you.

May I suggest a compromise? You don't want carpet because it holds dust and pollen to aggavate your son's condition. Your downstairs neighbor wants something on the floor to quiet the noise. Have you considered rubber flooring? The type that goes together like jigsaw puzzle pieces. They use this in daycare centers, garages, and offices with cement floors. You can get it in home improvement centers. Since it is rubber and has no open pores, it doesn't hold allergins. It can be mopped to keep it clean. It comes in many colors to be pleasing to the kids. It muffles sound. Check this out as it may be a good compromise.

Dear Sir/ Madam,
I thank you for your reply and suggestion.
I have considered it and bought some at costco last month to put in the kids bedroom. The good - it mops up, the bad - all liquids and some solids go through the seams of the pieces and my kids have already figured out that they can pull them up and take them apart. My nieghbor refuses to open the door and speak with me. He makes me feel that he wants dead silence. Case in point my kids arrived this morning at 8:30am. At 9am deafening bass music was blasting from his apartment - my kids were running around happy to be home. They got scared when they heard that loud noise. At 10:30am they went to sleep (after an overseas flight). at 11:15am the music stopped. As of writing this reply they are still sleeping and its quiet.
I also considered mediation, but that only works if both parties are willing to listen.

Any other thoughts?

He's not going to get dead silence. Not from a downstairs apartment, and not even from a single family house. There is always some noise. The key is to moderate the noise the noise to normal levels. Consider putting a runner of the same material in the halls and through the living room. Call the police when there is deafening music from below. You need to prove that he is disturbing you too. Get copies of the police reports. Make a list of what steps you have taken to minimize noise to the tenant below and send them to the LL. Send copies of any police report of the neighbor's music to the LL to show that it is a two way street. Let the LL know that you are trying your best to not disturb the tenant but that children will make normal noise. He cannot refuse to rent to people with children as it would be discriminatory. Good luck.