TITLE: How nilpotent is the ring of stable homotopy groups of spheres?
QUESTION [5 upvotes]: Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi_r^S(S^0)$?

REPLY [3 votes]: If one uses the Adams-Spectral sequence based on cohmology theories other than $BP$ it is possible to say a little more. In "A vanishing line in the $BP \langle 1 \rangle$-Adams spectral sequence" Jesús González shows that when $p$ is odd the $E_2$-page of the $BP\langle 1 \rangle$-Adams spectral sequence for the sphere spectrum has a vanishing line of slope $(p^2-p-1)^{-1}$. This is used to show that the $E_\infty$-term of the classic ASS has a vanishing line of slope $(2p-1)/[(2p-2)(p^2-p-1)]$. The result you are looking for is then Corollary 4.4:


In the $n$-th stable stem, any $p$-local homotopy class not detected by the $J$-spectrum is annihilated by $p^{\text{min}\{4,p\}+ \epsilon_n+S(n)}$, where $q=2p-2$, $S(n)$ is the first integer larger than or equal to $(q+1)n/[q(p^2-p-1)]$ and $\epsilon_n = 0$ unless $n+2$ is divisible by $q$, in which case $\epsilon_n$ is the highest power of $p$ dividing $n+2$.