In a recent paper, Chan, Laba, and Pramanik investigated geometric configurations inside thin subsets of the Euclidean set possessing measures with Fourier decay properties. In this paper we ask which configurations can be found inside thin sets of a given Hausdorff dimension without any additional assumptions on the structure. We prove that if the Hausdorff dimension of E ⊂ R d , d ≥ 2, is greater than d+1 2 , then for each k ∈ Z + there exists a non-empty interval I such that, given any sequence {t 1 , t 2 , . . . , t k ; t j ∈ I}, there exists a sequence of distinct points {x j } k+1 j=1 , such that x j ∈ E and |x i+1 − x i | = t j , 1 ≤ i ≤ k. In other words, E contains vertices of a chain of arbitrary length with prescribed gaps.
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 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 generalized 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.
We prove that if E ⊂ R 2d , for d ≥ 2, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by dim H (E), and φ is a sufficiently regular function, then the upper Minkowski dimension of the setin line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.
A theorem of Steinhaus states that if E ⊂ R d has positive Lebesgue measure, then the difference set E − E contains a neighborhood of 0. Similarly, if E merely has Hausdorff dimension dim H (E) > (d + 1)/2, a result of Mattila and Sjölin states that the distance set ∆(E) ⊂ R contains an open interval. In this work, we study such results from a general viewpoint, replacing E − E or ∆(E) with more general Φ -configurations for a class of Φ : R d × R d → R k , and showing that, under suitable lower bounds on dim H (E) and a regularity assumption on the family of generalized Radon transforms associated with Φ, it follows that the set ∆ Φ (E) of Φ-configurations in E has nonempty interior in R k . Further extensions hold for Φ -configurations generated by two sets, E and F , in spaces of possibly different dimensions and with suitable lower bounds on dim H (E) +
Abstract. We extend a result, due to Mattila and Sjölin, which says that if the Hausdorff dimension of a compact set2 , then the distance set ∆(E) = {|x − y| : x, y ∈ E} contains an interval. We prove this result for distance sets ∆ B (E) = { x − y B : x, y ∈ E}, where · B is the metric induced by the norm defined by a symmetric bounded convex body B with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.
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