Let $n \in \mathbb { N }$. Assuming the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ of zeros of $\varphi_n$ constructed in IV.D.1, deduce that the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ satisfies the asymptotic distribution property: $$\forall c \in ] 0,1 \left[ , \quad \exists j \in \mathbb { N } \quad \text { such that } \quad \forall k \in \mathbb { N } , 0 < \alpha _ { j + k + 1 } ^ { ( n ) } - \alpha _ { j + k } ^ { ( n ) } < \frac { \pi } { c } \right.$$
Let $n \in \mathbb { N }$. Assuming the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ of zeros of $\varphi_n$ constructed in IV.D.1, deduce that the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ satisfies the asymptotic distribution property:
$$\forall c \in ] 0,1 \left[ , \quad \exists j \in \mathbb { N } \quad \text { such that } \quad \forall k \in \mathbb { N } , 0 < \alpha _ { j + k + 1 } ^ { ( n ) } - \alpha _ { j + k } ^ { ( n ) } < \frac { \pi } { c } \right.$$