Additive basis of order $2$ (I)

Question

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

Update : I crossposted to MO.

Answer 1

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