A note on non-periodic greedy expansions in Salem base

Let \(\beta \) $\beta$ be a Salem number with \(\beta >1\) $\beta >1$ and x be an element in \({\mathbb {Q}}(\beta ) \cap [0,1)\) $\mathbb{Q}(\beta )\cap [0,1)$ . K. Schmidt conjectured that any element in \({\mathbb {Q}}(\beta ) \cap [0,1)\) $\mathbb{Q}(\beta )\cap [0,1)$ has a periodic greedy expansion in base \(\beta \) $\beta$ . In this note, we prove that if x has a non-periodic greedy expansion in base \(\beta \) $\beta$ then the sequence of the greedy expansion of x has higher complexity more than polynomial complexity.