Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$.

# Question

Given $\epsilon > 0$ (may be assumed to be very small), what is a reasonable **upper bound** for the tail probability $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2z_i^2 \ge \epsilon)$ ?

# Observations

Using ideas from this other answer (MO link), one can establish the

**non-uniform**anti-concentration bound: $P(\sum_{i=1}^na_i^2z_i^2 \le \epsilon) \le \sqrt{e\epsilon}$ for all $a \in \mathbb S_{n-1}$.The uniform analogue is another story. May be one can use covering numbers ?