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][1].  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][2] of a set of topological recurrence which is not a set of measurable recurrence.  Discovered [independently by Ruzsa][3].

- [Forrest's example][4] of a set of measurable recurrence which is not a set of strong recurrence (and [McCutcheon's variant][5] of Forrest's example).

 - [Green's version of niveau sets][1] (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][6] 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 (Unpublished, to my knowledge.)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][7] of sets which are $k$-Bohr recurrent but not $(k+1)$-Bohr recurrent.

- [Julia Wolf's construction][8] of sets whose popular difference sets lack structure.

- [My construction][9] 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][10] of the above to countable abelian groups.

- [My construction][11] 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.


  [1]: https://arxiv.org/pdf/math/0409420.pdf
  [2]: https://zbmath.org/?q=an%3A0641.05044
  [3]: https://sites.uml.edu/daniel-glasscock/files/2021/06/Ruzsa_difference_sets_1985.pdf
  [4]: https://zbmath.org/?q=an%3A0773.28014
  [5]: https://zbmath.org/?q=an%3A0867.28010
  [6]: https://zbmath.org/?q=an%3A0728.11009
  [7]: https://zbmath.org/?q=an%3A0981.05038
  [8]: https://zbmath.org/?q=an%3A1246.05026
  [9]: https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/abs/recurrence-rigidity-and-popular-differences/3EDBA68AD011C028C2310D82D0AED81E
  [10]: https://zbmath.org/?q=an%3A7481816
  [11]: https://discreteanalysisjournal.com/article/26859-separating-bohr-denseness-from-measurable-recurrence