Answers are given to two questions concerning the existence of some sparse subsets of $\mathscr{H} = \{0, 1, ..., H - 1\} \subseteq * \mathbb{N}$, where H is a hyperfinite integer. In $\S$ 1, we answer a question of Kanovei by showing that for a given cut U in $\mathscr{H}$, there exists a countably determined set $X \subseteq \mathscr{H}$ which contains exactly one element in each U-monad, if and only if $U = a \cdot \mathbb{N}$ for some $a \in \mathscr{H} \backslash \{0\}$. In $\S$2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut $V \subseteq \mathscr{H}$ such that for any cut U, (i) there exists a U-discrete set $X \subseteq \mathscr{H}$ with $X + X = \mathscr{H}$ (mod H) provided $U \subsetneqq V$, (ii) there does not exist any U-discrete set $X \subseteq \mathscr{H}$ with $X + X = \mathscr{H}$ (mod H) provided $\supsetneqq V$. We obtain some partial results for the case U = V.