Erdős similarity conjecture for rapidly decaying sequences

Determine whether the Erdős similarity conjecture holds for arbitrary rapidly decaying infinite sequences C = {a_n} ⊂ ℝ (for example, sequences with a_n ↓ 0 and a_{n+1}/a_n → 0): specifically, decide whether for every such C there exists a measurable set E ⊂ ℝ with positive Lebesgue measure that contains no affine copy of C (i.e., there do not exist x ∈ ℝ and t ∈ ℝ \ {0} with (x + tC) ⊂ E).

Background

The authors emphasize that prior results (Falconer and Eigen) cover slowly decaying sequences where a_{n+1}/a_n → 1, but do not address rapidly decaying sequences. In contrast, their constructions contain rapidly decaying sequences, yet the general status of the conjecture for such sequences remains unresolved.

They further show that their lattice-based constructions force additive branching leading to Bourgain’s non-universality criterion, producing specific positive-measure sets with the Erdős similarity property even in the presence of rapidly decaying sequences. Nonetheless, the general problem for rapidly decaying sequences is still open.