What image resolution can/should be used for the icon.png file for an iPhone app? I know the size is 57 x 57 but what about the resolution?
Resolution only matters when you are trying to match the size of an image across multiple devices (print, screen, etc)
App icons are always 57x57 and will display without any scaling thus resolution doesn't matter (you could save it as 72dpi, 65535dpi or missing the dpi metadata entirely--SpringBoard won't care and will draw it the same in all three cases)
Here is the link to Apple guidelines, with the sizes for iPad, iPhone retina, etc:
https://developer.apple.com/library/ios/documentation/userexperience/conceptual/mobilehig/AppIcons.html#//apple_ref/doc/uid/TP40006556-CH19-SW1
The "resolution" matters, but don't confuse "resolution" with dpi. Dpi is a measure used for printing, it means "dots per inch". It works like this: if you have an image of 100 x 100 saved at 100dpi, it will measure on paper exactly 1 inch (100 pixels per inch).
The dpi for the icons doesn't really matter, as they are intended for screen display only, not for printing.
It is customary to save images as 72dpi - the so called "screen resolution", because the old 14'' monitors (remember those?) could only display a maximum of 72 pixels per every inch of screen. This is no longer true, especially for the "retina" display of the iPhone. It has a much denser screen resolution, it can show much more pixels in a single inch than the older models of the iPhone. This is why you have to save 2 icons: one for retina display at 114 x 114 pixels, and another for the older iPhones, at 57 x 57. Again, the dpi doesn't matter, because whatever dpi you choose, the number of pixels on your image does not change: it would always have exactly 12,996 pixels (114 x 144).
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