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Interior of sums of planar sets and curves

Published online by Cambridge University Press:  05 September 2018

KÁROLY SIMON  and
KRYSTAL TAYLOR
KÁROLY SIMON
Affiliation:
Budapest University of Technology and Economics, Department of Stochastics, Institute of Mathematics, and MTA-BME Stochastics Research Group, 1521 Budapest, P.O. Box 91, Hungary e-mail: simonk@math.bme.hu
KRYSTAL TAYLOR
Affiliation:
Department of Mathematics, The Ohio State, Columbus, OH, 231 W. 18th Ave., MW 706 Columbus, OH 43210 e-mail: taylor.2952@osu.edu
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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 ${\mathcal C}^2$ 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 119 - 148
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

† This work came out of a collaboration that started at ICERM at Brown University, Rhode Island. This research was supported by the Mathematics Research Institute of the Ohio State University.

‡

Partially supported by the grant OTKA 123782, by MTA-BME Stochastics Research Group and by ICERM, by supporting his participation on one of their a semester programs in 2016.

References

REFERENCES

[1] Erdogan, B. A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Notes (2006).Google Scholar
[2] Erdős, P. and Oxtoby, J. Partitions of the plane into sets having positive measure in every non-null measurable product set. Trans. Amer. Math. Soc. 79 (1955), 91–102.Google Scholar
[3] Falconer, K. J. The geometry of fractal sets. Cambridge Tracts in Mathematics, 85 (Cambridge University Press, Cambridge, 1986).Google Scholar
[4] Hewett, D. P. and Moiola, A. On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space. Anal. Appl. 15, Issue 15 (2017).Google Scholar
[5] Iosevich, A., Mourgoglou, M. and Taylor, K. On the Mattila-Sjölin theorem for distance sets. Ann. Acad. Sci. Fenn. Math. 37, no. 2 (2012).Google Scholar
[6] Iosevich, A., Taylor, K. and Uriarte-Tuero, I. Pinned geometric configurations in Euclidean space and Riemannian manifolds. https://arxiv.org/pdf/1610.00349.pdf.Google Scholar
[7] Jàrai, A. Regularity properties of functional equations in several variables. Adv. Math. 8. (Springer, New York, 2005).Google Scholar
[8] Marstrand, J. M. Packing circles in the plane. Proc. London Math. Soc. 55 (1987), 37–58.Google Scholar
[9] Mattila, P. Geometry of sets and measures in Euclidean spaces. Stud. Adv. Math. vol. 44 (Cambridge University Press, Cambridge, 1995).Google Scholar
[10] Mattila, P. Fourier analysis and Hausdrff dimension, Cambridge University Press, Studies in Advanced Mathematics vol. 150, (2015). 2, 10, 32Google Scholar
[11] Mitsis, T. On a problem related to sphere and circle packing. J. London Math Soc. (2), (1999).Google Scholar
[12] Oberlin, D. Packing Spheres and Fractal Strichartz in ℝd for d ⩾ 3. Proc. Amer. Mth. Soc. 134, no. 11 (2006).Google Scholar
[13] Rudin, W. Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics) (McGraw-Hill Book co., 3rd ed., 1976).Google Scholar
[14] Palis, J. and Takens, F. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Stud. Adv. Math. vol. 35 (Cambridge University Press, 1993).Google Scholar
[15] Peres, Y. and Schlag, W. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102, (2000), no. 2, 193–251.Google Scholar
[16] Piccard, S. Sur les ensembles de distances des ensembles de points d'un espace Euclidien. Memoires de l'Universite de Neuchatel, vol. 13. (L'Universite de Neuchatel, 1939). 212 pages.Google Scholar
[17] Rams, M. and Simon, K. Projections of fractal percolations. Ergodic Theory Dynam. Syst. 35, 2, (2015), 530–545.Google Scholar
[18] Shmerkin, P. On the Hausdorff dimension of pinned distance sets. Preprint (2017).Google Scholar
[19] Simon, K. and Taylor, K. Dimension and measure of sums of planar sets and curves. Preprint (2017).Google Scholar
[20] Steen, A. L. and Seebach, A. J. Counterexamples in Topology. (Springer-Verlag, Berlin, New York, 1978).Google Scholar
[21] Steinhaus, H. Sur les distances des points des ensembles de mesure positive. Fund. Math. 1, (1920), 93–104.Google Scholar
[22] Stromberg, K. An elementary proof of Steinhaus' Theorem. Proc. Amer. Math. Soc. 36, (1972), no. 1, vol. 119, no. 5 (1997), 985–1026.Google Scholar
[23] Wolff, T. Lectures on harmonic analysis. Edited by Łaba and Carol Shubin. University Lecture Series, 29 (Amer. Math. Soc., Providence, RI, 2003).Google Scholar
[24] Wolff, T. Local smoothing type estimates on Lp for large p. Geom. Funct. Anal. vol. 10 (2000).Google Scholar