A recent paper proves that for any unbounded subset A of the real line ℝ, and for any small ε between 0 and 1, one can find a closed set S that occupies at least 1 − ε of every unit interval yet contains no nontrivial affine copy of A. This settles the large‑scale version of the Erdős similarity problem, a question asked by Erdős in 1974 about whether infinite sets always avoid being replicated in sets of positive measure.

The proof uses a probabilistic construction. By randomly deleting small intervals while preserving symmetry, the authors build a set that is (1 − ε)-large and yet lacks any affine copy (a scaled and translated version) of any unbounded set A. This is the first time such a universal largest‑measure set has been shown to exist for all unbounded A.

However, the rate at which the proportion of points inside a unit interval approaches 1 cannot be chosen independently of A. A probabilistic argument shows that if the set’s complement in each interval drops below a certain threshold, the set inevitably contains a scaled copy of a given infinite sequence. Thus the construction is constrained by the specific geometry of A.

These results advance the understanding of pattern avoidance in measurable sets and highlight limitations of large‑measure constructions.