The spectrum of the growth rate of the tunnel number is infinite

In a previous paper Kobayashi and Rieck defined the growth rate of the tunnel number of a knot $K$, a knot invariant that measures the asymptotic behavior of the tunnel number under iterated connected sum of $K$. We denote the growth rate by $\mbox{gr}_t(K)$. In this paper we construct, for any $\epsilon > 0$, a hyperbolic knots $K \subset S^{3}$ for which $1 - \epsilon < \mbox{gr}_t(K) < 1$. This is the first proof that the spectrum of the growth rate of the tunnel number is infinite.