Countable unions of intervals without Riesz spectra

We prove that every set $S = \bigcup_{n \geq 0} [ t_{2 n + 1} , t_{2 n} )$, where $t_0 > t_1 > \cdots > 0$ and $t_n \to 0$, admits no Riesz spectrum. This extends the result of Kozma, Nitzan, and Olevskii \cite{KNO} to a wide class of countable unions of intervals.