### 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 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})° = Ø., where S^{1} denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S^{1})° ≠ 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:= 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 Ø' 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.

Original language | English |
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Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

DOIs | |

Publication status | Accepted/In press - Jan 1 2018 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*. https://doi.org/10.1017/S0305004118000580