Post Undeleted by Vidit Nanda
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Start with $n = 2$. An element of $M(2,k)$

Okay, so this is determined by a single $nowhere near a \in {0,\ldots,k}~$ for $k \geq 2$. Since we wish to avoid $a=0,k~$ complete solution, but this is as far as I got and hopefully someone else sees it from here:

It is clear that $P(2,k) = \frac{k-1}{k+1}~$ which is certainly non-decreasing in $k$. Keep in mind the valid choices of $a$: these come from $[0,k] \cap \mathbb{Z}~$. Observe that $[0,k]$ is the standard simplex $S$ of side $k$ in dimension $1$ and that the $0$ entries correspond to the boundary of this simplex.

The case

When $n=3$ is much more interesting: n=3$, the matrices in$M(3,k)$correspond to four integers$a_{11},a_{12},a_{21},a_{22}~$from$0,\ldots,k~$such that • For$i = 1,2~$the sum 1$ or $2~$, $\sum_j a_{ij} \leq k$
• For $j = 1,2~$ the sum 1$or$2~$,$\sum_i a_{ij} \leq k$• We get These hyperplane inequalities carve out a zero entry if either one of the convex region$a_{ij}$is zero or if one of the inequalities above is an equality. Each of these situations corresponds to C \subset \mathbb{R}^4$ from the point with coordinates cube $a_{ij}$ lying on a boundary face of [0,k]^4$and the four dimensional "prism" which consists zero entry" cases of the cube$[0,k]^4$intersected with the half space from M(3,k)$ are precisely the third inequalitybounding faces of this region.

Sobottom line: I think , if a theorem establishes that the ratio $$\frac{\text{integral points on the boundary of this prism}}{\text} C}{\text{ the total number of integral points in this prism}}$$ }C} decreases as one increases $k$, which leads to then we obtain your desired result. The I don't know enough about convex polytopes to cite something here but it sounds reasonable just from dimension considerations...

Ideally, this process generalizes would generalize to higher dimensions. An element of $M(n,k)$ has zero entries if and only if the vector of entries in the first $(n-1) \times (n-1)$ block lies in the boundary of convex polytope carved from the cube $[0,1]^{(n-1)^2}$ by $2n-1$ hyperplanes.

Post Deleted by Vidit Nanda
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