More complete intersection theorems

The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a k-uniform t-intersecting family on n points, and describes all optimal families. In recent work, we extended this theorem to the weighted setting, giving the maximum μ_p measure of a t-intersecting family on n points. In this work, we prove two new complete intersection theorems. The first gives the supremum μ_p measure of a t-intersecting family on infinitely many points, and the second gives the maximum cardinality of a subset of Z_m ⁿ in which any two elements x,y have t positions i₁,…,i_t such that x_ij −y_ij ∈{−(s−1),…,s−1}. In both cases, we determine the extremal families, whenever possible.