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Configuration Sets with Nonempty Interior

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Abstract

A theorem of Steinhaus states that if \(E\subset \mathbb {R}^d\) has positive Lebesgue measure, then the difference set \(E-E\) contains a neighborhood of 0. Similarly, if E merely has Hausdorff dimension \(\hbox {dim}_{{\mathcal {H}}}(E)>(d+1)/2,\) a result of Mattila and Sjölin states that the distance set \(\varDelta (E)\subset \mathbb {R}\) contains an open interval. In this work, we study such results from a general viewpoint, replacing \(E-E\) or \(\varDelta (E)\) with more general \(\varPhi \)-configurations for a class of \(\varPhi :\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}^k,\) and showing that, under suitable lower bounds on \(\hbox {dim}_{{\mathcal {H}}}(E)\) and a regularity assumption on the family of generalized Radon transforms associated with \(\varPhi ,\) it follows that the set \(\varDelta _\varPhi (E)\) of \(\varPhi \)-configurations in E has nonempty interior in \(\mathbb {R}^k\). Further extensions hold for \(\varPhi \)-configurations generated by two sets, E and F,  in spaces of possibly different dimensions and with suitable lower bounds on \(\hbox {dim}_{{\mathcal {H}}}(E)+\hbox {dim}_{{\mathcal {H}}}(F).\)

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Notes

  1. In applications, one can easily localize away from singularities of \(\varPhi \) at degenerate configurations (at \(x=y\) for this \(\varPhi \)).

  2. We refer to the monographs of Mattila [25, 26] for background definitions and results.

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Authors and Affiliations

  1. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

    Allan Greenleaf & Alex Iosevich

  2. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Krystal Taylor

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  1. Allan Greenleaf
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  2. Alex Iosevich
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  3. Krystal Taylor
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Correspondence to Allan Greenleaf.

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Dedicated to the memory of Eli Stein, who inspired by his scholarship, teaching and friendship.

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Greenleaf, A., Iosevich, A. & Taylor, K. Configuration Sets with Nonempty Interior. J Geom Anal 31, 6662–6680 (2021). https://doi.org/10.1007/s12220-019-00288-y

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