Common null complements for polynomial images of finite sums of perfect sets

Fix an integer D ≥ 1, and let A₁, …, A₍D₊₂₎ be nonempty compact perfect subsets of the real line. For their Minkowski sum C = A₁ + ⋯ + A₍D₊₂₎, we construct a single null Fσ set B ⊆ ℝ such that P(C) + B = ℝ for every nonconstant real polynomial P of degree at most D. In particular, every sum of three compact perfect sets has one null complement that works under every nonzero dilation. Thus the conclusion of Jung and Lai’s Conjecture 4.8 holds for these sums; the conjecture remains open for arbitrary perfect sets. The conclusion applies to some sums with zero logarithmic Hausdorff and packing dimensions that contain no polynomial image of an infinite summable finite-digit product. The proof isolates a local sum hypothesis that permits the same construction for more general compact sets.

GitHub repository: https://github.com/eulogica/common-null-complements-perfect-sums