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In the Boolean lattice, Sperner's, Erd\H{o}s's, Kleitman's and Samotij'stheorems state that families that do not contain many chains must have a veryspecific layered structure. We show that if instead of $\mathbb{Z}_2^n$ we workin $\mathbb{Z}_{2^n}$, several analogous statements hold if one replaces theword $k$-chain by projective cube of dimension $2^{k-1}$.  We say that $B_d$ is a projective cube of dimension $d$ if there are numbers$a_1, a_2, \ldots, a_d$ such that $$B_d = \left\{\sum_{i\in I} a_i \bigg\rvert\emptyset \neq I\subseteq [d]\right\}.$$  As an analog of Sperner's and Erd\H{o}s's theorems, we show that whenever$d=2^{\ell}$ is a power of two, the largest $d$-cube free set in$\mathbb{Z}_{2^n}$ is the union of the largest $\ell$ layers. As an analog ofKleitman's theorem, Samotij and Sudakov asked whether among subsets of$\mathbb{Z}_{2^n}$ of given size $M$, the sets that minimize the number ofSchur triples (2-cubes) are those that are obtained by filling up the largestlayers consecutively. We prove the first non-trivial case where $M=2^{n-1}+1$,and conjecture that the analog of Samotij's theorem also holds.  Several open questions and conjectures are also given.

The largest projective cube-free subsets of $\mathbb{Z}_{2^n}$