What is the best possible $\epsilon$ and sequence $(a\_n)\_{n = 1}^\infty \subset [0, 1]$ we can find such that 
$$
d\_{N}:=\sup\_{x\in [0,1]}\inf\_{n=1}^N |x-a\_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in \mathbb{N}$? Note that dyadic decomposition provides $\epsilon=1$.



Sequence of nested sets in $[0, 1]$ with bound on gaps
Let me begin by reformulating the question a little bit, making it precise, and also avoiding what happens for the initial finite number of $N$. Given an infinite sequence $a\_n$ in $[0,1]$, for every $N$ let $u\_N$ denote the smallest gap between the numbers $a\_1$, $\ldots$, $a\_N$ (when arranged in ascending order between $0$ and $1$) and let $v\_N$ denote the maximum gap. The problem is essentially interested in understanding $v\_N/2$ (the point $x$ is halfway between the ends of the maximum gap). The problem as formulated above was studied by A. Ostrowski, [de Bruijn and Erdos](http://www.dwc.knaw.nl/DL/publications/PU00018611.pdf) and also solved (somewhat later) by [Toulmin](http://link.springer.com/article/10.1007/BF01899983#page-2). de Bruijn and Erdos (and Toulmin) showed that for any sequence one has 
$$ 
\limsup\_{N\to \infty} Nv\_N \ge\frac{1}{\log 2} = 1.4426\ldots, 
$$ 
and this value is attained for the sequence 
$$ 
a\_n = \{ \log\_2 (2n-1)\}, 
$$ 
where $\{ x\}$ stands for fractional part, and $\log\_2$ denotes the logarithm to base $2$. They also show that $\liminf\_{N\to \infty} Nu\_N$ is always at most $1/\log 4$ and this too is attained in the example above. You might also find interesting this related recent paper of [Chung and Graham](http://www.math.ucsd.edu/~ronspubs/16_01_discrepancy.pdf). 


My first thought on seeing the problem was sequences of the form $\{ n\theta\}$ for an irrational $\theta$. I think that in this class of examples, the golden ratio $\phi = (1+\sqrt{5})/2$ might be best and gives a normalized largest gap of $\phi^3/\sqrt{5} = 1.8944\ldots$ (if I calculated correctly). This is not as good as Toulmin's example. Nevertheless, you might also find interesting this survey of the [three gaps theorem](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.501.3023&rep=rep1&type=pdf) (which appeared in L'Enseignement). 


