Consider $C_a$ in $\Bbb Z^n$ a cube of height $a$ at origin in positive coordinates with one corner at origin.
Consider the set $M_c$ of all unimodular matrices in $\Bbb Z^{n\times n}$ with each entry bound in absolute value by $c>0$.
(1) If you pick two random matrices $R_1$ and $R_2$ in $M_{c+\epsilon}$ and transform all vectors in $C_{c-\epsilon}$ on average how many collisions can we expect?
(2) On average how many random matrices we should pick in $M_{c+\epsilon}$ so that when we transform each point in $C_{c-\epsilon}$ we will fill at least ${1-\frac1{n^\alpha}}$ of the cube $C_{c}$ where $\alpha>1$ holds?
(3) What is minimum number of matrices we should pick in $M_{c+\epsilon}$ so that when we transform each point in $C_{c-\epsilon}$ we will fill at least ${1-\frac1{n^\alpha}}$ of the cube $C_{c}$ where $\alpha>1$ holds?
