Balanced-density conjecture for p-adic rotational difference sets

Prove that for every odd prime p and every positive integer r, if A and B are subsets of (Z/p^rZ)^2 with equal densities satisfying δ_A = δ_B ≫ p^{-1}, then |gA − B| ≫ p^{2r} for a positive proportion of g ∈ SO_2(Z/p^rZ).

Background

The paper studies conditions under which the rotational difference set gA − B occupies a positive proportion of (Z/p^rZ)^2 for a positive proportion of rotations g in SO_2(Z/p^rZ). The main theorems establish sufficient density conditions in the unbalanced case: for p ≡ 3 (mod 4), δ_A^{1/2}δ_B ≥ 2p^{-1}, and for p ≡ 1 (mod 4), δ_Aδ_B ≥ 2p^{-1}.

Examples involving unions of cosets of the subgroup of vectors divisible by p, together with subsets of a p-adic circle, indicate that the exponent and density thresholds cannot generally be improved in the same way. The stated conjecture concerns the balanced case and would reach the natural density scale δ_A = δ_B ≫ p^{-1}. The authors note that it appears highly challenging and may be comparable in difficulty to the Erdős–Falconer distance problem.