In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32[![enter image description here][1]][1]

**Question:** When $p=2$, $k\geq 2$ and $n=\infty$, what is the dimension of each generator $j^*(x_I^*)$? I do not quite understand the argument.

The related context is in the following. 

[![enter image description here][2]][2]

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/LeTYn.png
  [2]: https://i.sstatic.net/9evFL.png
  [3]: https://i.sstatic.net/4hEsK.jpg