Configuration Sets with Nonempty Interior
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Configuration Sets with Nonempty Interior Allan Greenleaf1
· Alex Iosevich1 · Krystal Taylor2
Received: 29 July 2019 © Mathematica Josephina, Inc. 2019
Abstract A theorem of Steinhaus states that if E ⊂ Rd has positive Lebesgue measure, then the difference set E − E contains a neighborhood of 0. Similarly, if E merely has Hausdorff dimension dimH (E) > (d + 1)/2, a result of Mattila and Sjölin states that the distance set Δ(E) ⊂ R contains an open interval. In this work, we study such results from a general viewpoint, replacing E − E or Δ(E) with more general Φconfigurations for a class of Φ : Rd ×Rd → Rk , and showing that, under suitable lower bounds on dimH (E) and a regularity assumption on the family of generalized Radon transforms associated with Φ, it follows that the set ΔΦ (E) of Φ-configurations in E has nonempty interior in Rk . 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 dimH (E) + dimH (F). Keywords Configurations · Falconer distance problem · Thin sets · Fourier integral operators Mathematics Subject Classification 35S30 · 28A80 · 44A12
1 Introduction A classical theorem of Steinhaus [38] states that if E ⊂ Rd , d ≥ 1, with positive Lebesgue measure, |E|d > 0, then the difference set E − E ⊂ Rd contains a neigh-
Dedicated to the memory of Eli Stein, who inspired by his scholarship, teaching and friendship.
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Allan Greenleaf [email protected] Alex Iosevich [email protected] Krystal Taylor [email protected]
1
Department of Mathematics, University of Rochester, Rochester, NY 14627, USA
2
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
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A. Greenleaf et al.
borhood of the origin. E − E can interpreted as the set of two-point configurations, x − y, of points of E modulo the translation group. A variant of this was obtained by Mattila and Sjölin [27] for thin sets, i.e., E with |E|d = 0 but satisfying a lower bound on the Hausdorff dimension, dimH (E), in the context of the Falconer distance problem: if Δ(E) is the distance set of E, Δ(E) := {|x − y| : x, y ∈ E} ⊂ R, then if dimH (E) > d+1 2 , it follows that Δ(E) contains an open interval. The purpose of the current paper is to generalize these results in two ways: to two-point configurations in E as measured by a general class of Φ-configurations, which can be nontranslation-invariant, and indeed not even in Euclidean space, and to allow asymmetric configurations between sets in different spaces, e.g., between points and lines or points and circles in R2 , or lines and lines in R3 . In the process, we shall establish non-empty interior results for some configuration sets for which previously it was not even known that the configuration space has positive Lebesgue measure. In order to formulate these more general results, consider the models E − E and Δ(E) as the images of E × E under the maps (x, y) → x − y and (x, y) → |x − y|, resp. Now consider a C ∞ function Φ : Rd × Rd → R k , k ≤
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