I encountered the following value in my research:



> 
> Let $n,m$ be some integer. Suppose $\alpha\_1,\dots,\alpha\_m$ are unit vectors in $\mathbb{R}^n$.
>  Denote
>  $$
> L = \mathop{\mathrm{E}}\_x[ \prod\_{1\leq j\leq m} \langle \alpha\_j,x \rangle^2],
> $$
>  where $x \in \mathbb{R}^n$ is a random vector whose $\ell$th component is i.i.d. uniformly over $\{-1,1\}$.
> 
> 
> 


My observation is that if $m=n$, $L$ may be $0$. For instance, let $m=n=2$, $\alpha\_1=\frac{1}{\sqrt{2}}(1\ 1)$ and $\alpha\_2=\frac{1}{\sqrt{2}}(1\ {-1})$.
However, if, say, $m=o(n)$ or even $n/2$, then $L$ is *seemingly* lower bounded by $\Omega(1)$. How can I lower bound $L$ for those small $m$ (relative to $n$)?


For small and concrete $m$, I can manually lower bound $L$ by first expand the RHS and then apply the *random subsum principle* (namely, $\mathrm{E}\_x[x\_{\ell\_1}x\_{\ell\_2} \dots x\_{\ell\_c}]=0$, where $x\_{\ell\_j}$ denotes the $\ell$th component of $x$). But I am not sure how to generalize this approach to arbitrary $m$.


Any hint or reference will be greatly appreciated (let alone an answer).




---


For those who are curious about the background:
I am working on *quantum query complexity*, and I am trying to use the *polynomial method* to solve some certain problems.
If you are interested but not familiar with these terms, refer to [[BdW02]](https://www.sciencedirect.com/science/article/pii/S030439750100144X).



Lower bound of the expectation of the product of inner products of random vectors
The exact lower bound on $L$ is $0$ for any $n\ge2$ and $m\ge2$. Indeed, for $j=1,\dots, n$, let $a\_j:=\alpha\_j$ be any unit vectors in $\mathbb R^n$ such that $a\_1=\frac1{\sqrt{2}}(1,1,0,\dots,0)$ and $a\_2=\frac1{\sqrt{2}}(1,-1,0,\dots,0)$. Then 
$\prod\_{1\leq j\leq m} \langle \alpha\_j,x \rangle^2=0$ and hence $L=0$. 


