Word maps, polynomial maps and image ratios

If A is a finite group (or a finite ring) and is a word map (or a polynomial map), we define the quantity | (A) |/|A| as the image ratio of on A and will be denoted by (, A). In this article, we investigate the set R () =\ (, A): A is a finite group\, and also consider the case of rings. Specifically, we demonstrate the existence of word maps (and polynomial maps) whose set of image ratios is dense in 0, 1 for both groups (and rings).