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 ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ ∅. 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 := Cγ × Cγ, γ ⩾ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 − 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.

中文翻译：

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平面集和曲线之和的内部

最近，对两个平面集的算术和的研究受到了相当大的关注。我们专注于了解内部（一种+ Γ)°，当 Γ 是分段${\数学 C}^2$曲线和一种⊂ℝ2. 首先，我们给出一个非常大的示例（全测量、密集、Gδ） 放一种这样（一种+小号1)° = ∅，其中小号1表示单位圆。这表明，仅一种不保证（一种+小号1)° ≠ ∅。然而，如果我们假设一种是两个相当大的集合的一种广义乘积，则 (一种+ Γ)° ≠ ∅，只要 Γ 具有非零曲率。作为我们方法的副产品，我们证明了固定距离集C:=Cγ×Cγ, γ ⩾ 1/3, 固定在C有非空的内部，其中Cγ(见 (1.1)) 是中间的 1 − 2γ 康托尔集（包括通常的中间三分之一康托尔集，C1/3）。我们对中间三分之一康托集的证明需要一个单独的方法。我们也证明C+小号1有非空的内部。