Let $$\alpha = (1+\sqrt{5})/2$$ α = ( 1 + 5 ) / 2 be the golden ratio, and let $$B(\alpha ) = (\left\lfloor n\alpha \right\rfloor )_{n\ge 1}$$ B ( α ) = ( n α ) n ≥ 1 and $$B(\alpha ^2) = \left( \left\lfloor n\alpha ^2\right\rfloor \right) _{n\ge 1}$$ B ( α 2 ) = n α 2 n ≥ 1 be the lower and upper Wythoff sequences, respectively. In this article, we obtain a new estimate concerning the fractional part $$\{n\alpha \}$$ { n α } and study the sumsets associated with Wythoff sequences. For example, we show that every $$n\ge 4$$ n ≥ 4 can be written as a sum of two terms in $$B(\alpha )$$ B ( α ) and a positive integer n can be written as the sum $$\left\lfloor a\alpha \right\rfloor +\left\lfloor b\alpha ^2\right\rfloor $$ a α + b α 2 for some $$a, b\in {\mathbb {N}}$$ a , b ∈ N if and only if n is not one less than a Fibonacci number. The structure of the set $$B(\alpha ^2)+B(\alpha ^2)$$ B ( α 2 ) + B ( α 2 ) contains some kinds of fractal and palindromic patterns and is more complicated than the other sets, but we can also give a complete description of this set.

中文翻译：

---

与 Wythoff 序列和斐波那契数列相关的和集

令 $$\alpha = (1+\sqrt{5})/2$$ α = ( 1 + 5 ) / 2 为黄金比例，令 $$B(\alpha ) = (\left\lfloor n\ alpha \right\rfloor )_{n\ge 1}$$ B ( α ) = ( n α ) n ≥ 1 and $$B(\alpha ^2) = \left( \left\lfloor n\alpha ^2 \right\rfloor \right) _{n\ge 1}$$ B ( α 2 ) = n α 2 n ≥ 1 分别是下和上 Wythoff 序列。在本文中，我们获得了关于小数部分 $$\{n\alpha \}$$ { n α } 的新估计，并研究了与 Wythoff 序列相关的和集。例如，我们证明每个 $$n\ge 4$$ n ≥ 4 可以写成 $$B(\alpha )$$B ( α ) 中的两项之和，一个正整数 n 可以写成总和 $$\left\lfloor a\alpha \right\rfloor +\left\lfloor b\alpha ^2\right\rfloor $$ a α + b α 2 对于某些 $$a, b\in {\mathbb { N}}$$ a , b ∈ N 当且仅当 n 不小于斐波那契数。