This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this question are quite heavy and in fact unnecessary, so I omit the details for now. My knowledge of geometry and functional analysis is poor hence this post.
Consider the vector spaces $V=\mathbb{R}^N$. We are going to let $N\rightarrow\infty$ and pose a question about the size of a particular subset (as a function of $N$). Let $(u,v)$ denote the angle between two vectors so $\langle u,v\rangle=\|u\|\|v\|\cos(u,v)$.
For each $V$ let $S$ denote the subset comprising of those $s\in V$ for which $\|s\|\sim N^{1/2}$. Also let $M$ denote invertible linear operators and let $(b_n)\subset S$ be (non-orthogonal) bases for each $V$ (in fact $b_n=(M^{-1})^{*}e_n$ where $(e_n)$ are the standard orthonormal bases).
The following is true:
For every $s\in S$ we have $$\frac{\cos^2(Ms,b_n)}{n}>> N^{-3}$$ for a non-sparse subset $1\leq n\leq N$.
I am interested in those $s\in S$ which also have a particular property $P$ say, and it turns out that for this subset one also has:
If $s$ satisfies $P$ then $$\frac{\cos^2(Ms,b_n)}{n}<< N^{-3}$$ for all sufficiently large $n$.
In other words, the rate of approach of $(Ms,b_n)\rightarrow \pi/2$ is completely determined if $s$ has property $P$, and this rate is optimal. My questions are as follows:
Do these bounds together imply that those $s$ with property $P$ belong to subset that is asymptotically smaller than $S$ itself?
What geometric tools are used to conclude this is the case if so?
