Existence of some sparse sets of nonstandard natural numbers

Answers are given to two questions concerning the existence of some sparse subsets of K = {0.1..... H - 1} subset of or equal to N+. where H is a hype rfinitr integer. In 1. we answer a question of Kanovei by showing that For a given cut U in K, there exists a countably determined set X subset of or equal to K which contains exactly one element in each U-monad. if and only if U = a (.) N for some a is an element of K \ {0}. In 2, we deal with a qu estion of Keisler and Leth in [6]. We show that there is a cut V subset of or equal to K such that for any cut U. (i) there exists a U-discrete set X subset of or equal to K with X + X = K (mod H) provided U not subset of or equal to V. (ii) there does not exist any U-discrete set X subset of or equ al to K with X + X = K (mod H) provided U not superset of or equal to V. We obtain some partial results for the case U = V.