Simultaneous avoidance of power families of null sequences

Let (a_n) be a positive sequence tending to zero with liminf a_(n+1)/a_n > 0. For every η ∈ (0,1), we construct a compact set E ⊆ [0,1] of Lebesgue measure greater than 1−η such that {x+s a_n^λ : n ≥ 1} is not contained in E for any x ∈ ℝ, s ≠ 0, and λ > 0. No monotonicity is required, and one set works for all positive powers simultaneously, with infinitely many distinct points of every copy outside E. Taking a_n=e^(−n) gives one compact set of measure arbitrarily close to one containing no infinite geometric progression of any ratio in (0,1), answering negatively a question of Burgin, Goldberg, Keleti, MacMahon and Wang. We also prove a uniform finite hitting statement for ratios in a compact range and scales in [1,2]. The proof adapts random routing to grids shared by all exponents, with a localized count of exponent and scale parameters and a compactness repair for exceptional centres.

GitHub repository: https://github.com/eulogica/power-family-avoidance