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.

  • $\begingroup$ I believe it has been proved that any $\alpha\le5/4$ (perhaps slightly larger) works. $\endgroup$ – Henri Cohen Jul 8 '18 at 14:04
  • $\begingroup$ Do you have a reference about this result ? $\endgroup$ – uvdose Jul 8 '18 at 15:41
  • $\begingroup$ 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. $\endgroup$ – Henri Cohen Jul 10 '18 at 21:30
  • $\begingroup$ Thank you for your reply. I will try to find an article. $\endgroup$ – 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:

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


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