[EDIT: rambling class p rapids "stream of conciousness" largely removed]
Many ergodic theorists have been interested in Penrose tilings and the theory of aperiodic tiling spaces generally. One place where this theory intersects at least two of the topics you mentioned (Fourier analysis and difference sets) is in the application of Meyer sets; see R. V. Moody, "Meyer Sets and Their Duals", The Mathematics of Long-Range Aperiodic Order
, Kluwer, 1997. In this paper, Moody establishes seven equivalent characterizations of Meyer sets (a generalization of lattice suitable for a kind of harmonic analysis) is that a relatively dense
set [itex]\Latex \subset R^n[/itex] is a Meyer set iff the difference set [itex]\Lambda-\Lambda[/itex] is uniformly discrete
. Here, [itex]\Lambda[/itex] is relatively dense if there is a compact set such that [itex]R^n = K + \Lambda[/itex]. So taking n=1 and K=[0,1], the integers are relatively dense in R. And [itex]\Lambda[/itex] is uniformly discrete if there is an open neighborhood of the origin, U, such that the difference set misses U.
I am thinking of the example you gave as a "patch" from a "tiling" with prototiles of length 1,2,4, namely
In the rambling first version of this post, I seized upon translation invariants aspects of your comments and ignored everything else. This prompts me to inquire whether one can arbitrage
sum sets versus difference sets?
I've always been intrigued by Moody's paper but AFAIK this point of view has not been followed up, and it occurs to me that AFAIK Szemeredi's theorem has not been applied directly to tiling theory.
I think that I'm trying to suggest that it might be suggestive to try to interpret additive phenomena in the integers using some of the language of tiling theory, which might suggest some interesting problems. In addition, while IMO the general theory of tiling has not yet appeared, I expect it should provide a scheme for founding mathematics upon tilings rather than upon sets. If so from this POV it would not be surprising that "additive phenomena" can describe seemingly unrelated phenomenon.
The discovery of Penrose tilings gave rise to a great deal of interest in how unexpectedly rigid long range order can result from simple local rules. One could say that Szemeredi theorem concerns unexpectedly unavoidable order of a kind. So there does seem to be at least a vague spiritual connection.
Incidently, Bombieri, who worked on analytic topics related to some of the topics you mentioned, was one of those who became intrigued by Penrose tilings!