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 ${\mu }_p$ measure of a $t$-intersecting family on $n$ points. In this work, we prove two new complete intersection theorems. The first gives the supremum ${\mu }_p$ measure of a $t$-intersecting family on infinitely many points, and the second gives the maximum cardinality of a subset of ${\mathbb{Z}}_m^n$ in which any two elements $x,y$ have $t$ positions $i_1,\dots ,i_t$ such that $x_{i_j}?y_{i_j}\in \{?(s?1),\dots ,s?1\}$. In both cases, we determine the extremal families, whenever possible.