The structure of difference sets in additive combinatorics provides a curious example of this phenomenon. A specific instance is the following: unless specifically constructed to be a counterexample, if $A$ is a subset of $(\mathbb Z/2\mathbb Z)^d$ with $|A|> 0.01\cdot 2^d$ then the difference set $A - A:=\{a-a':a, a'\in A\}$ must contain a subgroup of index $K$ (independent of $d$ or $A$). The counterexamples, due independently to Igor Kriz and Imre Ruzsa, are spelled out explicitly in Theorem 9.4 of Ben Green's Finite Field Models in Arithmetic Combinatorics. Such constructions are often referred to as niveau sets.

What's curious is that niveau sets are, in some sense, the only known way to construct a dense subset $A$ of an abelian group $G$ where $A-A$ lacks some prescribed structure. Here are the instances I am aware of:

• Kriz's construction of a set of topological recurrence which is not a set of measurable recurrence. Discovered independently by Ruzsa.

• Forrest's example of a set of measurable recurrence which is not a set of strong recurrence (and McCutcheon's variant of Forrest's example).

• Green's version of niveau sets (Theorem 9.4): $A\subset (\mathbb Z/2\mathbb Z)^d$ where $|A|\approx (1/4)2^d$ and $A-A$ does not contain a subgroup of small index.

• Ruzsa's construction of dense sets $A\subset \{1,\dots,N\}$ where $A+A$ does not contain exceptionally long arithmetic progressions.

• Bourgain's example of subsets in $\mathbb T^d$ with Haar measure $m(A)\approx 1/2$ where $A-A$ does not contain a connected subgroup of $\mathbb T^d$. (Unpublished, to my knowledge.)

• Katznelson's examples of sets which are $k$-Bohr recurrent but not $(k+1)$-Bohr recurrent.

• Julia Wolf's construction of sets whose popular difference sets lack structure.

• My construction of a set $S\subset \mathbb Z$ where every translate of $S$ is a set of measurable recurrence and no translate of $S$ is a set of strong measurable recurrence.

• Ackelsberg's generalization of the above to countable abelian groups.

• My construction of a set dense in the Bohr topology of $\mathbb Z$ which is not a set of measurable recurrence.

While varying in many technical details, all of the above examples rely, in the same way, on the additive structure of Hamming balls in $(\mathbb Z/p\mathbb Z)^d$ for a fixed prime $p$ (usually $p=2$) and large $d$.

It would be very interesting to find a fundamentally different construction of a set $A$ where $A-A$ lacks some prescribed structure, or to prove that every such example comes from niveau sets.