TITLE: Additive basis of order 2
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: For $1<\alpha<\frac32$, $(\lfloor n^{\alpha}\rfloor)_{n\geqslant0}$ is an asymptotic basis of order 2. I finally found these two articles:
J-M. Deshouillers, Un problème binaire en théorie additive, Acta Arith. 25 (1974), 393-403
S.V. Konyagin, An additive problem with fractional powers, Mathematical Notes, 2003, 73:4, 594–597