Give an example of a set $\mathcal{X}$ of $N$ points for which no scaled projection onto a subspace of dimension $m \ll \log{N}$ is an approximate isometry. Context and hint: this is exercise 5.3.4 from Vershynin's book (HDP). Link to text (https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-b...

I have the following argument which I encountered, and can't seem to find why it's not true: Let $X$ be a topological space and let $A$ and $B$ be nonempty subsets of $X$. Then $\overline{A\cap B}^A=\overline{A\cap B}^X\cap A$. "Proof" $\overline{A\cap B}^A =\cap \{ F\subset A: \; F\supseteq A\ca...

Let $a$ be a nonnegative integer. For a given positive integer $k$, let $b_1,b_2,\ldots,b_k$ be odd integers greater than $1$. Using this result, it can be shown that, for each integer $n$, $$f_{a;b_1,b_2,\ldots,b_k}(n):=n^a\,\prod_{i=1}^k\,\left(n^{b_i}-n\right)$$ is divisible by $$\Gamma\left...