I just purchased an iPad Air for my daughter to use when she returns to Grad school this fall. We have an iPad to and the 6 mm broad-tip styluses worked OK for quick notes. With the higher resolution screen and a few years to improve things, I thought that then would have gotten better.
In fact, I think that they have gotten worse. We purchased a Defender case for the Air and I bought several of the inexpensive rubber tipped styluses. They all required too much pressure for writing comfortably and all skipped leaving the writing to be unusable.
Next, we purchased the 'Pencil' from 53 the makers of the iPad app 'Paper'. This uses a low powered bluetooth tech that WITHOUT the screen protector worked much better. You could use it to take notes using the Goodnote App. With the screen protector, the skipping returns and the pressure causes the tip to rub on the screen too much.
I have ordered the Adonit Script to try that, but there are several posts on the net that suggest that I won't be pleased with that either if using a screen protector. Honestly, does anyone really take their Air into an environment with people jostling about and NOT protect it?
I went to the Apple Store and their answer was "We don't sell a stylus, so we don't know anything about them." I made an appointment with the local Genius who admitted that they (the local store genuses were aware of the problem, but that Apple had not official response to the problem. [Seems that a problem doesn't exist if you don't acknowledge it.]
Our plan is to wait for the Adonit Jot Script stylus arrival, and if it doesn't work, we will return the Air and look at alternatives.
Hope you have better luck or better yet find the answer so that the rest of us can use the Air to its full potential.