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The paper considers the input-constrained binary erasure channel (BEC) withcausal, noiseless feedback. The channel input sequence respects the$(d,\infty)$-runlength limited (RLL) constraint, i.e., any pair of successive$1$s must be separated by at least $d$ $0$s. We derive upper and lower boundson the feedback capacity of this channel, for all $d\geq 1$, given by:$\max\limits_{\delta \in [0,\frac{1}{d+1}]}R(\delta) \leqC^{\text{fb}}_{(d\infty)}(\epsilon) \leq \max\limits_{\delta \in[0,\frac{1}{1+d\epsilon}]}R(\delta)$, where the function $R(\delta) =\frac{h_b(\delta)}{d\delta + \frac{1}{1-\epsilon}}$, with $\epsilon\in [0,1]$denoting the channel erasure probability, and $h_b(\cdot)$ being the binaryentropy function. We note that our bounds are tight for the case when $d=1$(see Sabag et al. (2016)), and, in addition, we demonstrate that for the casewhen $d=2$, the feedback capacity is equal to the capacity with non-causalknowledge of erasures, for $\epsilon \in [0,1-\frac{1}{2\log(3/2)}]$. For$d>1$, our bounds differ from the non-causal capacities (which serve as upperbounds on the feedback capacity) derived in Peled et al. (2019) in only thedomains of maximization. The approach in this paper follows Sabag et al.(2017), by deriving single-letter bounds on the feedback capacity, based onoutput distributions supported on a finite $Q$-graph, which is a directed graphwith edges labelled by output symbols.

Bounds on the Feedback Capacity of the $(d,\infty)$-RLL Input-Constrained Binary Erasure Channel