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Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in the $n$-line of the $E_2$ of the ASS of the sphere?
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element $...