Someone wants to use my app name

I have n app on the App Store it doesn't doing terribly well because I am neglecting it a little but I intend to go back to it soon.

Basically a big gaming company has approached me and wanted to buy the name. I have said I am not interest in the amount they are offering and told them my terms.

They don't seem to be budging and have asked me to sign a mutual agreement to co exist. If I refuse to sign will they still be able to use the name?

Do I have any rights ? Also if they do manage to use the name which is exactly the same as mine but with a space between will it benefit me if they are driving downloads? Also if they do well can they later try to get mine removed ?

It's the trademark issue, not copyright issue. If you have not file the trademark for your app name, you stand no chance. It's a good idea to explore the terms with that buyer. But, you need to find an attorney specializing in trademark/copyright services.

You did not mention which App Store is this (iOS / Android) - but I will assume it is the iOS one for now.

On the iOS App Store - 2 apps cannot exist with exactly the same name. Apple does not allow it. So, the second app must modify the name (Google does not care about this - so this is not an issue on the Android store). On iOS, people generally get around this by adding a space or an 's' in their name. For example, you can have a game called the "Word Game" and this company could do either a "Words Game" or maybe a "WordGame" or even "The Word Game".

Update: As pointed out by Geoff in the comments below, you can actually create a new app with the same app name as an existing app if the original developer has not set the meta data in multiple languages (I was not aware of this). Please follow the link in the comment to know more. All of this is allowed and you cannot do much about it unless you have trademarked your name and REALLY want to protect it for some reason.

To answer your questions:

If I refuse to sign will they still be able to use the name?

Yes. They will. They are only asking you to sign because they can protect themselves later if you decide to file a complaint with Apple or sue.

Also if they do manage to use the name which is exactly the same as
mine but with a space between will it benefit me if they are driving
downloads?

Yes. This had actually happened to us.
We had written a game almost 3 years ago and forgotten about it. It was actually a whole new category of Trivia Games and we had the best name. But our app wasn't great and we hadn't spent a lot of time on it. Lets call our app: Car Quiz.

The new app that was published a year ago was called: Cars Quiz (they did not ask us for permission). But the other game became a hit and still features in the top recommended games on the iOS. As a result, it drove quite a bit of downloads for us and helped us go back to the app, improve it and make some money.

Even though the new game was very similar in concept to ours - we did not do anything (in my experience - it is just too painful and expensive to do these things). We just improved monetization on our own game and benefited from free publicity. :)

I know we'll never get to making the same money as the new guy - but atleast we got something out of a forgotten project for free. It also helped shape our company in the direction that we took.

Also if they do well can they later try to get mine removed ?

No. They cannot. In my experience with both Apple, Google and Windows Stores, they cannot do this.
If you were there first, and unless there is a law suit in court, Apple, Google or Microsoft really don't care about this stuff.

My Advice

My advice would be to take a hard look at your app and see if it has potential to make money. As in - do you monetize well, is your app free / freemium / paid. Are people already paying for it?

If you can work on it quickly and make it a better monetized app, you should do it as soon as possible - because the big gaming company will spend a lot of money in promotions which will drive up your downloads as well. This has no meaning if you don't make any money out of your app. The promotions won't last for ever - so you have a short window. (You may as well sell the name and make some money).

It can also be that there is not much potential to monetize your app.
For example, there are many apps which people will use for free - but not spend money on it. If your goal is to make money, and you cannot monetize your app, then it might be a good idea to sell the name.

Else, fix it and enjoy the ride :)

Do let us know what you finally decide on.