Abstract
We prove analogues of the Szemerédi–Trotter theorem and other incidence theorems using \(\delta \)-tubes in place of straight lines, assuming that the \(\delta \)-tubes are well spaced in a strong sense.
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J. Bourgain. The discretized sum-product and projection theorems. Journal d’Analyse Mathématique, (1)112 (2010), 193–236
K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir and E. Welzl. Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom., 5 (1990), 99–160
A. Cordoba. The Kakeya maximal function and the spherical summation multipliers. Amer. J. Math., (1)99 (1977), 1–22.
C. Demeter, L. Guth, and H. Wang. Small cap decouplings. In: arXiv:1908.09166 (2019)
Gy. Elekes and M. Sharir. Incidences in three dimensions and distinct distances in the plane. In: Proceedings 26th ACM Symposium on Computational Geometry (2010), pp. 413–422
K. J. Falconer. On the Hausdorff dimensions of distance sets. Mathematika, (2)32 (1985), 206–212 (1986)
L. Guth, A. Iosevich, Y. Ou and H. Wang. On Falconer’s distance set problem in the plane. In: arXiv:1808.09346 (2018).
L. Guth, N. H. Katz. On the Erdős distinct distance problem in the plane. Ann. of Math. (2), (1)181 (2015), 155–190
T. Keleti and P. Shmerkin. New bounds on the dimensions of planar distance sets. In arXiv:1801.08745 (2018)
P. Mattila. Spherical averages of Fourier transforms of measures with finite energy: dimensions of intersections and distance sets. Mathematika, 34 (1987), 207–228
N. Katz and T. Tao. Some connections between Falconer’s distance set conjecture and sets of Furstenburg type. New York J. Math., 7 (2001), 149–187
T. Orponen. On the distance sets of Ahlfors–David regular sets. Adv. Math., 307 (2017), 1029–1045
T. Orponen. On the dimension and smoothness of radial projections. In: arXiv:1710.11053v2 (2017)
P. Shmerkin. On distance sets, box-counting and Ahlfors regular sets. Discrete Anal. 22 (2017)
P. Shmerkin. On the Hausdorff dimension of pinned distance sets. Iseral Journal of Mathematics, (2)230 (2019), 942–972
T. Wolff. Recent work connected with the Kakeya problem. Prospects in mathematics (Princeton, NJ, 1996), 129-162, Amer. Math. Soc., Providence, RI, 1999
T. Wolff. A Kakeya-type problem for circles. Amer. J. Math. (5)119 (1997), 985–1026
T. Wolff. Decay of circular means of Fourier transforms of measures. Int. Math. Res. Not. (10)(1999), 547–567
T. Wolff. Local smoothing type estimates on \(L^p\) for large p. Geom. Funct. Anal. (5)10 (2000), 1237–1288
Acknowledgements
The first author is supported by a Simons Investigator Award. We would like to thank Misha Rudnev for the very nice observation that simplifies the proof of Theorem 1.2. We would like to thank the anonymous referee for a careful reading of the draft and the many helpful suggestions that have improved the exposition of the article. Also, thanks to Yuqiu Fu, Shengwen Gan, Dominique Maldague, and Lingxian Zhang for helpful comments about a draft of the paper.
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Guth, L., Solomon, N. & Wang, H. Incidence Estimates for Well Spaced Tubes. Geom. Funct. Anal. 29, 1844–1863 (2019). https://doi.org/10.1007/s00039-019-00519-y
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DOI: https://doi.org/10.1007/s00039-019-00519-y