  We study closure properties for the Littlestone and threshold dimensions of
binary hypothesis classes. Given classes $\mathcal{H}_1, \ldots, \mathcal{H}_k$
of Boolean functions with bounded Littlestone (respectively, threshold)
dimension, we establish an upper bound on the Littlestone (respectively,
threshold) dimension of the class defined by applying an arbitrary binary
aggregation rule to $\mathcal{H}_1, \ldots, \mathcal{H}_k$. We also show that
our upper bounds are nearly tight. Our upper bounds give an exponential (in
$k$) improvement upon analogous bounds shown by Alon et al. (COLT 2020), thus
answering a question posed by their work.
