Let ${}\_2\pi\_n^S$ denote the $2$-power torsion subgroup of $n$th stable homotopy group of the sphere spectrum. Its order is a power of $2$: $$|{}\_2\pi\_n^S|=2^{k\_n}.$$ 


**Question:** What is known about growth of $k\_n$? Is it polynomial? What is the best estimation? 


**Remark:** Of course, any estimation on ${\rm Ext}^{s,t}\_{\mathcal A\_2}(\mathbb F\_2,\mathbb F\_2)$ implies an estimation on $k\_n.$ The Lambda algebra gives an exponential estimation, it is not interesting. I'm not very familiar with the May spectral sequence. Does it give a better estimation? 



Growth of stable homotopy groups of spheres
There is work by [Boedigheimer and Henn](https://link.springer.com/content/pdf/10.1007%2FBF01171748.pdf) that bounds the size of *unstable* homotopy groups of spheres or rather of the number of $p$-local summands (i.e. the dimension after tensoring with $\mathbb{F}\_p$). The bound is again exponential, namely $3^{q-n/2}$ for $\mathrm{dim}\_{\mathbb{F}\_p}\pi\_q(S^n)\otimes \mathbb{F}\_p$. There is a slight improvement in later work by [Henn](https://link.springer.com/content/pdf/10.1007/BF01172158.pdf), but the bound is still exponential as I understand it. 


Looking at the data, the growth of the stable homotopy groups seems to be less than exponential though. According to [Isaksen's charts](https://arxiv.org/pdf/1401.4983.pdf) (with possible miscounts by myself) the sequence of the first few $k\_n$ is:


1 1 3 0 0 1 4 2 3 1 3 0 0 2 6 2 4 4 4 3 2 2 8 2 2 2 3 1 0 1 8 4 5 5
 5 1 2 3 9 7 5 5 3 3 7 4 10 


Particularly big ones are $k\_{15} = 6$, $k\_{23} = 8$ and $k\_{47} = 10$. The contribution of the image of $J$ is $5$, $4$ and $5$ respectively in these degrees. While the image of $J$ should dominate in low degrees, elements of higher Adams-Novikov filtration should become more and more dominant. All in all, the data does not really look like an exponentially growing sequence, but who knows with our limited knowledge?


Edit: I incorporated Allen Hatcher's corrections to my sequence of $k\_n$. 


