On the distribution of the number of ones in a Boolean Pascal's triangle

This research is devoted to estimating the number of Boolean Pascal's triangles of large enough size $s$ containing a given number of ones $\xi\le ks$, $k>0$. We demonstrate that any such Pascal's triangle contains a zero triangle whose size differs from $s$ by at most constant depending only on $k$. We prove that there is a monotone unbounded sequence of rational numbers $0=k_0<k_1<\dotsc$ such that the distribution of the number of triangles is concentrated in some neighbourhoods of the points $k_is$. The form of the distribution in each neighbourhood depends not on $s$ but on the residue of $s$ some modulo depending on $i\ge 0$.