5 problems

Let $P_d$ be the probability that a random $0/1$-matrix of size $(d+1)\times(d+1)$ is singular; equivalently, in the associated $\pm1$ model, let $P_d$ denote the probability that…

For integers $m,n$ with $(m,n)\neq(1,1)$, let $f_{m,n}(k)=mn-\frac{k(m+n-k)mn}{(m+n)^2}-\left\{\frac{km}{m+n}\right\}\left\{\frac{kn}{m+n}\right\}$, where $\{x\}$ denotes the fract…

Let $\text{Sing}(m,n)$ denote the set of singular $m\times n$ real matrices, and write $d=m+n$ and $\delta_{m,n}=mn\left(1-\frac{1}{m+n}\right)$. KKLM's dimension conjecture. For a…

Hausdorff-dimension conjecture. The Hausdorff dimension of the set of singular $n\times m$ real matrices is equal to

Let $A$ be chosen uniformly from the set of $n\times n$ matrices with entries in $\{-1,+1\}$. Equivalently, $A\in\{-1,+1\}^{[n]^2}$, and singularity means $\det(A)=0$ in…