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http://static.howstuffworks.com/gif/iphone-11.gif
I should point out that this is a diagram of a resistive touchscreen - AFAIK the nano uses the apple preferred capacitive multitouch version instead.
This means; the layers don't flex and there is no need for a spacer layer (the 'dots').

That's not to say the pattern couldn't be in the touch digitizer. I haven't seen it, so I don't know. Besides the touch layer, and the LCD itself; another possibility is the backlight layers (reflective and diffusing, could have a regular pattern, or some light interference phenomenon.)
 
This is because the iPod Nano/iPhone screen has a tempered glass cover; most laptops + monitors do not for weight-saving reasons.

No, my point is that despite having a hard cover that should prevent pressure being put on the LCD screen my previous sixth gen would still let pressure get to the LCD screen just by pressing down enough to use the clip.

The new one doesn't seem so sensitive.
 
I know for a fact that all iPhone models, most if not all Android phones, all iPod touch models, and the new iPod nano use capacitive touchscreens, which do not have dots or a spacer layer. They also do not flex, as they have too thick a layer of glass to flex. I only saw the new iPod nano in person once (at an Apple store), and I did not notice dots, but I wasn't actually looking for them.

If there are any visible dots then I have no idea what they are for, but it's certainly not a resistive digitizer. The reason people have mentioned that Nintendo DS and DSi models have dots is because they use resistive digitizers.
 
I had an iPhone 3G and never noticed these. A few months ago, I broke the digitizer and replaced it with a cheap option from eBay. With the replacement, I could see a grid on the screen. I assumed it was because the replacement digitizer was of lower quality.

I now have an iPhone 4, and in certain angles, I can see a faint grid pattern on this screen as well (much less visible than on the cheap ebay digitizer).

Maybe your dots are related to this grid and have to do with the digitizer on the new Nanos?
 
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