Erdős similarity conjecture (general case)

Establish the Erdős similarity conjecture in full generality: for every infinite set C ⊂ ℝ, construct a measurable set E ⊂ ℝ with positive Lebesgue measure such that E contains no affine copy of C; equivalently, there do not exist x ∈ ℝ and t ∈ ℝ \ {0} with (x + tC) ⊂ E.

Background

The paper introduces the Erdős similarity problem and recalls that it asks whether for every infinite set C one can find a positive-measure set E that avoids all affine copies of C. The authors note that, despite progress in special regimes, the conjecture has not been settled in general.

They highlight partial advances: Bourgain’s criterion implying non-universality for sets containing a triple sum A+A+A, Kolountzakis’s almost-sure non-universality result, and recent progress for sets C with large logarithmic dimensions. Nevertheless, the overall conjecture remains unresolved.

References

The Erd\H{o}s similarity problem is a classical question in geometric combinatorics. It asks whether for every infinite set $C$ there is a measurable subset of the real line of positive measure so that the set does not contain an affine copy of $C$. This problem was raised by Erd\H{o}s in the 1970s. See . Although the conjecture is wide open, there are some results that are worth mentioning.

— Falconer lattice sets and the Erdos similarity problem  (2604.01493 - Iosevich et al., 2 Apr 2026) in Section 1 (Introduction)