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
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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](https://mathoverflow.net/a/95108/78539) (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 ?



Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$
As pointed out by a user (Nate Eldgredge) in the comments under the question, 


$$
P\left(\sup\_{a \in \mathbb S\_{n-1}}\sum\_{i=1}^n a\_i^2Z\_i^2 \ge \epsilon\right) = P\left(\max\_{1 \le i \le n}Z\_i^2 \ge \epsilon\right) = 1-P(Z\_1^2 \le \epsilon)^n = 1-P(\chi^2 \le \epsilon)^n.
$$
One can then invoke standard tail bounds for chi-squared variables (e.g exponential-type tail bound established in [Lemma 1 of this paper](https://projecteuclid.org/download/pdf_1/euclid.aos/1015957395)), to finish the problem.


