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Planar Euclidean analogue of the finite-field rigid-motions union bound

Establish a two-dimensional Euclidean analogue of the finite-field theorem for unions under rigid motions: for Borel sets E ⊂ R^2 and Θ ⊂ E(2) = O(2) × R^2, derive explicit quantitative conditions on E and Θ under which the Lebesgue measure of Θ(E) = {g(x) + z : (g, z) ∈ Θ, x ∈ E} is bounded below (in particular, is positive), in the spirit of the finite-field result asserting that if |E|^{1/2}·|Θ| ≫ q^3 then |Θ(E)| ≫ q^2 for E ⊂ F_q^2 and Θ ⊂ O(2) × F_q^2.

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Background

The paper presents finite-field analogues of packing results for unions under rigid motions, including a planar theorem (Theorem 1.6 here denoted thm_FF_dim2) that provides a lower bound on the size of the union Θ(E) in F_q2 under a product-type size condition on E and Θ. The authors supply alternative proofs leveraging ideas from their Euclidean results.

They explicitly note the absence of a counterpart in the Euclidean plane R2, indicating an unresolved problem to formulate and prove a quantitative planar theorem ensuring positivity or lower bounds for the Lebesgue measure of Θ(E) under natural conditions on E and Θ.

References

We provide simple alternative proofs of these results using some of the ideas from our ${\mathbb R}d$ results. However, we do not currently have an ${\mathbb R}2$ variant of Theorem \ref{thm_FF_dim2}. We hope to address this issue in the sequel.

— Packing sets in Euclidean space by affine transformations (2405.03087 - Iosevich et al., 6 May 2024) in Introduction, subsection “Union of sets by rigid motions in finite fields”