  The paper considers the input-constrained binary erasure channel (BEC) with
causal, 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 bounds
on the feedback capacity of this channel, for all $d\geq 1$, given by:
$\max\limits_{\delta \in [0,\frac{1}{d+1}]}R(\delta) \leq
C^{\text{fb}}_{(d\infty)}(\epsilon) \leq \max\limits_{\delta \in
[0,\frac{1}{1+d\epsilon}]}R(\delta)$, 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 binary
entropy 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 case
when $d=2$, the feedback capacity is equal to the capacity with non-causal
knowledge 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 upper
bounds on the feedback capacity) derived in Peled et al. (2019) in only the
domains of maximization. The approach in this paper follows Sabag et al.
(2017), by deriving single-letter bounds on the feedback capacity, based on
output distributions supported on a finite $Q$-graph, which is a directed graph
with edges labelled by output symbols.
