I'm investigating an idea to bound the norm of a random matrix on the unit sphere $\mathbb{S}^{d-1}$. I'm at a point where I need to bound the number of hypercubes of a $\epsilon$-fine grid intersecting the unit sphere. Unfortunately I've no background in geometry nor can I acquire one in time. Since I'm not even able to give a proper definition of what I mean, I drew an image of the number of cubes in $\mathbb{R}^2$ I want to count:
It would be great if anyone could give me some references if there are theorems in the literature.
Well...every cubie that intersects your sphere (in $n$-space, so I'm talking about $S^{n-1}$) contains a point at distance 1 from the origin...so the CLOSEST point to the origin in that cubie must be at least distance 1-D, where $D$ is the greatest distance between points of the cube...which happens to be $e \sqrt{n}$, where $e$ is the edge-length of your cubie. The same argument goes for how far OUTSIDE the sphere it can be.
Summary: every point of every cubie lies between the spheres of radii $$ r_1 = 1 - e \sqrt{n} \\ r_2 = 1 + e \sqrt{n} $$ From this, you can compute the volume of the inner and outer spheres (a bit of a mess, but Wikipedia will help you), and then take the difference of the two volumes and call it $U$. The total volume of your $k$ cubies will be
$k e^n$, and you know that $k e^n \le U$, hence $$ k \le \frac{U}{e^n} $$
So once you did a little work looking up formulas for sphere-volumes, you'll have an answer.
Let's just do it for $\mathbb R^2$ as shown above. In this case, we have \begin{align} r_{1,2} = 1 \pm e \sqrt{2} \end{align}
and the area of a disk of radius $r_1$ is $$ A_1 = \pi (r_1)^2 $$ and similarly for $r_2$, hence the difference is \begin{align} U &= \pi (r_2^2 - r_1^2) \\ U &= \pi (r_1 + r_2)\cdot (r_1 - r_2) \\ U &= \pi (2)\cdot (2e\sqrt{2}) \\ U &= 4\pi e\sqrt{2} \\ \end{align} Hence $$ k \le \frac{4 \pi e \sqrt{2}}{e^2} = \frac{4 \pi \sqrt{2}}{e}. $$
That actually sounds pretty reasonable: the count of cubies grows linearly in the inverse-length. If you try to work out the same formulas in 3-space, there'll be a difference-of-cubes in the numerator, which will be quadratic in $e$, and the denominator will be cubic in e, so you'll get the same general result. But there'll be extra terms, so the constant next to the $1/e$ will change with dimension in general.