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Here is a heuristic that agrees with the power proposed by @Bullet51 in the comments above, showing that $C(p)$ should grow like $p^{3/2}$. The sum should look like $pn$ times the typical order of the …

If $n$ points are placed uniformly at random in the unit square, then the distribution is very close to a Poisson process with intensity $n$. Scaling the process by $\sqrt n$, it’s like a Poisson proc …

The problem is essentially equivalent and slightly more symmetric if you make the rectangular surface "wrap around". (I'm assuming you want to have $A,B\gg r_c$?) You can of course also scale the prob …

Yes. What you are looking for follows from the known error bounds in the Gauss Circle Problem. In particular, in the notation of that Wiki article, $\text{card}(A_r\cap [a,b])=N(r+b)-N(r+a^-)$, while …

You're essentially asking if you drop $k$ objects each into one of $k$ bins, what is the probability that only $k/2$ bins are occupied. (You can forget the extra dimension and only think about the dim …

It doesn't make any difference that $d\gg 1$. If $0<a<b<1$, $Q(a,b)=\|a\mathbf x+(b-a)\mathbf y+(1-b)\mathbf z\|_2^2=a^2\|\mathbf x\|^2+(b-a)^2\|\mathbb y\|^2+(1-b)^2\|\mathbf z\|^2+2a(b-a)\langle\mat …