Some infinite elements in the Adams spectral sequence for the sphere spectrum

In the stable homotopy group πpnq+(p+1)q−1(V (1)) of the SmithToda spectrum V (1), the author constructed an essential element n for n ≥ 3 at the prime greater than three. Let β∗ s ∈ [V (1), S]spq+(s−1)q−2 denote the dual of the generator β′′ s ∈ πs(p+1)q(V (1)), which defines the β-element βs. In this paper, the author shows that the composite α1β1ξs ∈ πpnq+(s+1)pq+sq−6(S) for 1 < s < p − 2 is non-trivial, where ξs = β ∗ s−1 n ∈ πpnq+spq+(s−1)q−3(S) and q = 2(p − 1). As a corollary, ξs, α1ξs and β1ξs are also non-trivial for 1 < s < p − 2.