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 ξ ≤ 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 < ... such that the distribution of the number of triangles is concentrated in some neighbourhoods of the points ki s. The form of the distribution in each neighbourhood depends not on s but on the residue of s some modulo depending on i ≥ 0.