  Two dimensional field theories invariant under the Bondi-Metzner-Sachs (BMS)
group are conjectured to be dual to asymptotically flat spacetimes in three
dimensions. In this paper, we continue our investigations of the modular
properties of these field theories. In particular, we focus on the BMS torus
one-point function. We use two different methods to arrive at expressions for
asymptotic structure constants for general states in the theory utilising
modular properties of the torus one-point function. We then concentrate on the
BMS highest weight representation, and derive a host of new results, the most
important of which is the BMS torus block. In a particular limit of large
weights, we derive the leading and sub-leading pieces of the BMS torus block,
which we then use to rederive an expression for the asymptotic structure
constants for BMS primaries. Finally, we perform a bulk computation of a probe
scalar in the background of a flatspace cosmological solution based on the
geodesic approximation to reproduce our field theoretic results.
