  The article addresses some open questions about the relations between the
topological weak mixing property and the transitivity of the map $f\times f^2
\times...\times f^m$, where $f\colon X\ra X$ is a topological dynamical system
on a compact metric space. The theorem stating that a weakly mixing and
strongly transitive system is $\Delta$-transitive is extended to a
non-invertible case with a simple proof. Two examples are constructed,
answering the questions posed by Moothathu [Colloq. Math. 120 (2010), no. 1,
127--138]. The first one is a multi-transitive non weakly mixing system, and
the second one is a weakly mixing non multi-transitive system. The examples are
special spacing shifts. The later shows that the assumption of minimality in
the Multiple Recurrence Theorem can not be replaced by weak mixing.
