  In this paper, we continue a line of work on obtaining succinct population
protocols for Presburger-definable predicates. More specifically, we focus on
threshold predicates. These are predicates of the form $n\ge d$, where $n$ is a
free variable and $d$ is a constant.
  For every $d$, we establish a 1-aware population protocol for this predicate
with $\log_2 d + \min\{e, z\} + O(1)$ states, where $e$ (resp., $z$) is the
number of $1$'s (resp., $0$'s) in the binary representation of $d$ (resp., $d -
1$). This improves upon an upper bound $4\log_2 d + O(1)$ due to Blondin et al.
We also show that any 1-aware protocol for our problem must have at least
$\log_2(d)$ states. This improves upon a lower bound $\log_3 d$ due to Blondin
et al.
