Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?

I can think of something ad hoc like a Cantor middle $\frac13$ set where the middle $\frac13$ is followed by middle $\frac35$, middle $\frac57$ etc. but I am looking for something natural that's been studied before.

Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?

I can think of something ad hoc like a Cantor middle $\frac13$ set where the middle $\frac13$ is followed by middle $\frac24$, middle $\frac35$ etc. but I am looking for something natural that's been studied before.

Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?

I can think of something ad hoc like a Cantor middle-third set where the middle-thirds are followed by middle-5ths, middle-7ths etc. but I am looking for something natural that's been studied before.

Gratuitous illustration:

Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?

I can think of something ad hoc like a Cantor middle $\frac13$ set where the middle $\frac13$ is followed by middle $\frac35$, middle $\frac57$ etc. but I am looking for something natural that's been studied before.