The Furstenberg set and its random version

We study some number -theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers S = { 2(j) 3(k)} 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 uniformly distributed while T is, also that S is a Delta (p)-set for all 2 < p < infinity 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.