Using estimates for incomplete character sums with polynomial arguments, we provide a uniform distribution result for the dominating sets in Paley tournaments  Moreover, if k and q are fixed positive integers with q odd, then we construct a class of oriented regular graphs that are k-paradoxical (provided p is large enough) and “sparse” (with  edges) subgraphs of Paley tournaments  satisfying a similar uniform distribution property for the dominating sets.

tournaments, dominating sets, character sums, uniform distribution.