# Additive basis of order 2

Can we find $\alpha>1$ such that $u=(\lfloor n^\alpha\rfloor)_{n\geqslant0}$ is an additive basis of order $2$ (i.e. $\forall x\in\mathbb{N}, \exists(n,m)\in\mathbb{N}^2, x=u_n+u_m$) ?

Remark : This question has been asked previously on math.SE, but no response has been provided yet.

• I believe it has been proved that any $\alpha\le5/4$ (perhaps slightly larger) works. – Henri Cohen Jul 8 '18 at 14:04
• Do you have a reference about this result ? – uvdose Jul 8 '18 at 15:41
• Work of Jean-Marc Deshouillers around 1975-1978, but there is more recent stuff. I can give you the precise ref if you do not find it. – Henri Cohen Jul 10 '18 at 21:30
• Thank you for your reply. I will try to find an article. – uvdose Jul 10 '18 at 21:56

For $1<\alpha<\frac32$, $(\lfloor n^{\alpha}\rfloor)_{n\geqslant0}$ is an asymptotic basis of order 2. I finally found these two articles: