Then the maximal possible number of elements in $E?$

Question

Let $U$ be an orthonormal set in a Hilbert space $H$ and let $x \in H$ be such that $\vert \vert x \vert \vert =2$. Consider the set

$$E=\{ u\in U: \vert \langle x, u \rangle \vert \geq \frac{1}{4} \}$$

Then the maximal possible number of elements in $E?$

I use Bessel inequality but didn;t get any result. Please help.

Answer 1

Let $u_1,...,u_n \in E$ such that $u_j \ne u_k$ for $j \ne k$. Then

$$ \sum_{k=1}^n\vert \langle x, u_n \rangle \vert^2 \ge \frac{n}{16}.$$

On the other side, by Bessel:

$$ \sum_{k=1}^n\vert \langle x, u_n \rangle \vert^2 \le ||x||^2=4.$$

Can you proceed ?