Kneser's theorem for codes and $\mathrm{\ell }$-divisible set families

A $k$-wise $\mathrm{\ell }$-divisible set family is a collection $\mathcal{F}$ of subsets of $\{1,\dots ,n\}$ such that any intersection of $k$ sets in $\mathcal{F}$ has cardinality divisible by $\mathrm{\ell }$. If $k=\mathrm{\ell }=2$, it is well-known that $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le 2^{\lfloor n\mathrm{/}2\rfloor }$. We generalise this by proving that $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le 2^{\lfloor n\mathrm{/}p\rfloor }$ if $k=\mathrm{\ell }=p$, for any prime number $p$. For arbitrary values of $\mathrm{\ell }$, we prove that $4{\mathrm{\ell }}^2$-wise $\mathrm{\ell }$-divisible set families $\mathcal{F}$ satisfy $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le 2^{\lfloor n\mathrm{/}\mathrm{\ell }\rfloor }$ and that the only families achieving the upper bound are atomic, meaning that they consist of all the unions of disjoint subsets of size $\mathrm{\ell }$. This improves upon a recent result by Gishboliner, Sudakov and Timon, that arrived at the same conclusion for $k$-wise $\mathrm{\ell }$-divisible families, with values of $k$ that behave exponentially in $\mathrm{\ell }$. Our techniques rely heavily upon a coding-theory analogue of Kneser’s Theorem from additive combinatorics.