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- W2964223712 abstract "Abstract 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 ${mathcal C}^2$ curve and A ⊂ ℝ 2 . To begin, we give an example of a very large (full-measure, dense, G δ ) set A such that ( A + S 1 )° = ∅, where S 1 denotes the unit circle. This suggests that merely the size of A does not guarantee that ( A + S 1 )° ≠ ∅. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then ( A + Γ)° ≠ ∅ whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C := C γ × C γ , γ ⩾ 1/3, pinned at any point of C has non-empty interior, where C γ (see (1.1)) is the middle 1 − 2γ Cantor set (including the usual middle-third Cantor set, C 1/3 ). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S 1 has non-empty interior." @default.
- W2964223712 created "2019-07-30" @default.
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- W2964223712 date "2018-09-05" @default.
- W2964223712 modified "2023-09-25" @default.
- W2964223712 title "Interior of sums of planar sets and curves" @default.
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- W2964223712 doi "https://doi.org/10.1017/s0305004118000580" @default.
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