Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points (k,m)p^−m with 0 ≤ k ≤ n < p^m and (nk)≡a (mod⁡p)s$\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s $ (for (a, p) = 1) for m→∞ tends to the Hausdorff measure on the “p-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.