Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise curve and A ⊂ R2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = Ø., where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ Ø. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then (A + Γ)° ≠ Ø. whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C:= Cγ × Cγ, γ ≥ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 Ø' 2γ Cantor set (including the usual middle-third Cantor set, C1/3). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S1 has non-empty interior.
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|Publication status||Accepted/In press - Jan 1 2018|
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