  We study some number-theoretic, ergodic and harmonic analysis properties of
the Furstenberg set of integers $S=\{2^{m}3^{n}\}$ and compare them to those of
its random analogue $T$. In this half-expository work, we show for example that
$S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$
is, and that $S$ is a $\Lambda(p)$ set for all $2<p<\infty$ and that $T$ is a
$p$-Rider set for all $p$ such that $4/3<p<2$. Measure-theoretic and
probabilistic techniques, notably martingales, play an important role in this
work.
