Sumsets associated with Wythoff sequences and Fibonacci numbers

Abstract

Let $\alpha = (1+\sqrt{5})/2$ be the golden ratio, and let $B(\alpha ) = (\left\lfloor n\alpha \right\rfloor )_{n\ge 1}$ and $B(\alpha ^2) = \left( \left\lfloor n\alpha ^2\right\rfloor \right) _{n\ge 1}$ 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 $n\ge 4$ can be written as a sum of two terms in $B(\alpha )$ and a positive integer n can be written as the sum $\left\lfloor a\alpha \right\rfloor +\left\lfloor b\alpha ^2\right\rfloor$ for some $a, b\in {\mathbb {N}}$ 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.