Direct Sum Fails for Zero Error Average Communication

direct sum fails for average communication complexity
as well as for external information cost.
Our example also refutes
a version of a conjecture by Braverman et al that
in the zero error case amortized communication complexity equals external information cost.

In our examples the underlying distributions do not have full support.
One interpretation of a distributions of non full support is as a promise
given to the players (the players have a guarantee on their inputs).
This brings up the issue of promise versus non-promise problems
in this context.

Joint work with Gillat Kol, Amir Shpilka and Amir Yehudayoff