and A. Salch in [@thhmay]. The idea of the spectral sequence is to mimic the construction of May where he filters a Hopf algebra by powers of the augmentation ideal and constructs an associated filtration of the bar construction. In particular, the author shows in [@thhmay] that there is a model for the Whitehead tower of a connective commutative ring spectrum which produces a multiplicative filtration of that commutative ring spectrum. From this filtered ring spectrum, the author and A. Salch produce a filtration of a generalized bar construction, for example the cyclic bar construction, and construct a spectral sequence in higher topological Hochschild homology. Here we only use the version for the cyclic bar construction and this work is summarized in Section \[section 2\].

In the 1980’s, F. Waldhausen first suggested computing algebraic K-theory of connective versions of the localizations of the sphere spectrum appearing in the chromatic tower [@MR764579]. Recall that there are homology theories $E(n)$, depending on a positive integer $n$ and a prime $p$, called the Johnson-Wilson $E$-theories. Using Bousfield localization we can construct spectra $L_{E(n)}S_{(p)}$ which closely approximate the $p$-local sphere spectrum for large $n$ in the sense that the map $$S_{(p)} \lra \underset{\leftarrow}{\holim}\hspace{.05in} L_{E(n)}S_{(p)}$$ 