Define the distance between a point $x$ and a finite set $X$ to be $ d(x,X) :=\ \displaystyle\min_{y\in X} \left\{\ d\left( x , y\right)\ \right\}.$
Let $\ (x_n)_n\subset [0,1]\ $ be a real sequence, and for all $\ n\geq 2,\ $ let $\ y_n:= d\left(\ x_n, \{x_k: 1\leq k\leq n-1\}\ \right).$
Proposition: $\ \sum y_n\ $ diverges $\ \implies\ \{x_n:n\in\mathbb{N}\}\ $ is somewhere dense in $[0,1].$
And what about the converse of the proposition, that is:
$\ \sum y_n\ $ converges $\ \implies\ \{x_n:n\in\mathbb{N}\}\ $ is nowhere dense in $[0,1]\ ?$
The idea of the first proposition is that, if $\ \sum y_n\ $ diverges, then $y_n$ is a "large set", and so $x_{n+1}$ is "far away" from members of $(x_k)_{k\leq n}$, slowly "filling up" an interval in $[0,1].$ So $(x_n)$ is dense.
Any ideas for how to prove or disprove these propositions, or ideas for constructing counter-examples?