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Consider the set $$ \{\lambda = \alpha _1^2 + \alpha _2^2 + \alpha _3 ^2 : \alpha _j \in \mathbb{Z}, \, j=1,2,3 \} $$ of real numbers. Assume I order this set so that $\lambda _1 \le \lambda _2 \le \lambda _3 \le \cdots $ in increasing order. Of course every $\lambda _j$ will be repeated a few times (except for 0). I want to obtain a bound of the form $$ \lambda _j \le \varphi (j) $$
for some function $\varphi $. It probably doesn't have to be a sharp bound but the sharper the better, of course. At least I'm looking for a polynomial bound of the form $\lambda _j \le Cj^k$ for some positive integer $k$.

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The number of sequence elements $\lambda_j$ that are less than some $x$ is at least $(2\sqrt{x/3}-1)^3$, by considering $\alpha_1,\alpha_2,\alpha_3\in[-\lfloor\sqrt{x/3}\rfloor,\lfloor\sqrt{x/3}\rfloor]$. Therefore $\lambda_{\lceil(2\sqrt{x/3}-1)^3\rceil} \le x$, which implies that $\lambda_j \le Cj^{2/3}$ for some constant $C>0$.

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  • $\begingroup$ Just what I was looking for! Thank you so much! $\endgroup$ – flavio Dec 6 '13 at 14:27

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