The Furstenberg set and its random version

Abstract

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 $\Lambda (p)$-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.