The Furstenberg set and its random version

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