the only stage in this tower that has been completely computed, after $p$-completion inside, is mod $p$ homotopy of $K(\mathbb{Z}_p)$ due to work of S. Tsalidis [@MR1428060] and M. Bökstedt and I. Madsen [@MR1317117]. The purpose of this paper is to compute a linear approximation to the next stage in the tower after $p$-completion, that is we compute $THH((\tau_{\ge 0}L_{E(1)}S)_p)$ mod $(p,v_1)$ using the fact that $(\tau_{\ge 0}L_{E(1)}S)_p\simeq \tau_{\ge 0}L_{K(1)}S$, where $\tau_{\ge 0}L_{K(1)}S$ is notation for the connective cover of the $K(1)$-local sphere.

Algebraic K-theory of $\tau_{\ge 0}L_{K(1)}S$ is also modeled by iterated algebraic K-theory of certain finite fields, which can be explained as follows. For this explanation, let $p$ be an odd prime and let $q$ be a prime power that also topologically generates $\mathbb{Z}_p^{\times}$. Due to D. Quillen [@MR0315016], it is known that algebraic K-theory of a finite field of order $q$ can be computed, after $p$-completion, using the fiber sequence $$\xymatrix{ K(\mathbb{F}_q)_p \ar[r] & ku_p \ar[r]^{1-\psi_q} & \Sigma^{2}ku_p}$$ $\psi_q$ is the $q$-th Adams operation and $ku_p$ is the $p$-completion of connective complex K-theory. Now, E. Devinatz and M. Hopkins showed that there is an equivalence $$L_{K(1)}S \simeq KU_p^{h\mathbb{G}_1}$$ where $\mathbb{G}_1$ is the height $1$ Morava stabilizer group, 