Sumsets associated with Wythoff sequences and Fibonacci numbers

Let alpha=(1+5)/2alpha = (1+\sqrt{5})/2$$\end{document} be the golden ratio, and let B(alpha)=(n alpha)n >= 1 = (\left\lfloor n\alpha \right\rfloor )_{n\ge 1}$$\end{document} and B(alpha 2)=n alpha 2n >= 1 = \left( \left\lfloor n\alpha <^>2\right\rfloor \right) _{n\ge 1}$$\end{document} be the lower and upper Wythoff sequences, respectively. In this article, we obtain a new estimate concerning the fractional part {n alpha} and study the sumsets associated with Wythoff sequences. For example, we show that every n >= 4 can be written as a sum of two terms in B(alpha) and a positive integer n can be written as the sum a alpha+b alpha 2 for some a,b is an element of 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) 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.