Given $E \subset [0,1]$, is there a name for the following quantity, with $p$ integer :
$$\delta(p) = \sup \left \{ \sum \limits_{k=1}^n \sqrt{a_i-a_{i-1}}\ |\ n \ge 1,\ a_0<\cdot \cdot \cdot <a_n \mbox{ in } E,\ \forall i \in [\![1,n]\!],\ a_i - a_{i-1} \ge \frac{1}{p} \right \}$$
The closest concept I can think of is Hausdorff dimension, but it is not the same, as I assume $a_i-a_{i-1} \ge \frac{1}{p}$ and not $\le$.
In a similar fashion, I am also interested in the following quantity $$N(p) = \sup \left \{ n \in \mathbb{N}\ |\ \exists a_1,...,a_n \in E : \ \forall i \in [\![1,n-1]\!],\ a_{i+1}-a_i \ge \frac{1}{p} \right \}$$
For instance if $E$ is closed and with positive Lebesgue measure, I proved that $p = O\big(N(p)\big)$, but I don't know how to describe the behaviour of $N(p)$ for a general set (or rather, I can, but not in an explicit enough way). Hence I am looking for related concepts which could help characterizing precisely those two quantities.