Almost isoperimetric subsets of the discrete cube

We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + ε)$ can be made into a subcube by at most $(2 ε/\log_{2}(1/ε))|A|$ additions and deletions, provided $ε$ is less than an absolute constant. We deduce that if $A \subset \{0,1\}^{n}$ has size $2^{t}$ for some $t \in \mathbb{N}$, and cannot be made into a subcube by fewer than $δ|A|$ additions and deletions, then its edge-boundary has size at least $|A| \log_{2}(2^{n}/|A|) + |A| δ\log_{2}(1/δ) = 2^{t}(n-t+δ\log_{2}(1/δ))$, provided $δ$ is less than an absolute constant. This is sharp whenever $δ= 1/2^{j}$ for some $j \in \{1,2,\ldots,t\}$.

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