We prove two related concentration inequalities concerning the number of rational
points of hyperelliptic curves over subsets of a finite field. In particular, we investigate
the probability of a large discrepancy between the numbers of quadratic
residues and nonresidues in the image of such subsets over uniformly random
hyperelliptic curves of given degrees. We find a constant probability of such a
high difference and show the existence of sets with an exceptionally large
discrepancy.
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