Geometric incidence theorems via Fourier analysis

Iosevich, Alex; Jorati, Hadi; Łaba, Izabella

doi:10.1090/S0002-9947-09-04866-1

Geometric incidence theorems via Fourier analysis
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by Alex Iosevich, Hadi Jorati and Izabella Łaba PDF

Trans. Amer. Math. Soc. 361 (2009), 6595-6611 Request permission

Abstract:

We show that every non-trivial Sobolev bound for generalized Radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in Euclidean space. This mechanism allows us to deduce geometric results not readily accessible by combinatorial methods.

References

Boris Aronov and Micha Sharir, Cutting circles into pseudo-segments and improved bounds for incidences, Discrete Comput. Geom. 28 (2002), no. 4, 475–490. Discrete and computational geometry and graph drawing (Columbia, SC, 2001). MR 1949895, DOI 10.1007/s00454-001-0084-1
Boris Aronov, Vladlen Koltun, and Micha Sharir, Incidences between points and circles in three and higher dimensions, Discrete Comput. Geom. 33 (2005), no. 2, 185–206. MR 2121292, DOI 10.1007/s00454-004-1111-9
Boris Aronov, János Pach, Micha Sharir, and Gábor Tardos, Distinct distances in three and higher dimensions, Combin. Probab. Comput. 13 (2004), no. 3, 283–293. MR 2056399, DOI 10.1017/S0963548304006091
Michael Christ, Quasi-extremals for a Radon-like transform, preprint, 2006.
Kenneth L. Clarkson, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, and Emo Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), no. 2, 99–160. MR 1032370, DOI 10.1007/BF02187783
G. Elekes, Cs. Tóth: Incidences of not too degenerate hyperplanes, preprint, 2005.
K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), no. 2, 206–212 (1986). MR 834490, DOI 10.1112/S0025579300010998
Allan Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981), no. 4, 519–537. MR 620265, DOI 10.1512/iumj.1981.30.30043
Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
A. Iosevich, Fourier analysis and geometric combinatorics. To appear in the Birkhauser volume dedicated to the annual Padova lectures in analysis, 2004.
S. Hofmann and A. Iosevich, Circular averages and Falconer/Erdös distance conjecture in the plane for random metrics, Proc. Amer. Math. Soc. 133 (2005), no. 1, 133–143. MR 2085162, DOI 10.1090/S0002-9939-04-07603-8
A. Iosevich and I. Łaba, $K$-distance sets, Falconer conjecture, and discrete analogs, Integers 5 (2005), no. 2, A8, 11. MR 2192086
Alex Iosevich and Michael Rudnev, Distance measures for well-distributed sets, Discrete Comput. Geom. 38 (2007), no. 1, 61–80. MR 2322116, DOI 10.1007/s00454-007-1316-9
A. Iosevich and E. Sawyer, Maximal averages over surfaces, Adv. Math. 132 (1997), no. 1, 46–119. MR 1488239, DOI 10.1006/aima.1997.1678
Izabella Laba and József Solymosi, Incidence theorems for pseudoflats, Discrete Comput. Geom. 37 (2007), no. 2, 163–174. MR 2295051, DOI 10.1007/s00454-006-1279-2
János Pach and Micha Sharir, On the number of incidences between points and curves, Combin. Probab. Comput. 7 (1998), no. 1, 121–127. MR 1611057, DOI 10.1017/S0963548397003192
János Pach (ed.), Towards a theory of geometric graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004. MR 2065247, DOI 10.1090/conm/342
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157 (1986), no. 1-2, 99–157. MR 857680, DOI 10.1007/BF02392592
W. Schlag, On continuum incidence problems related to harmonic analysis, J. Funct. Anal. 201 (2003), no. 2, 480–521. MR 1986697, DOI 10.1016/S0022-1236(03)00081-8
Micha Sharir, On joints in arrangements of lines in space and related problems, J. Combin. Theory Ser. A 67 (1994), no. 1, 89–99. MR 1280600, DOI 10.1016/0097-3165(94)90005-1
Micha Sharir and Emo Welzl, Point-line incidences in space, Combin. Probab. Comput. 13 (2004), no. 2, 203–220. MR 2047237, DOI 10.1017/S0963548303005935
Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579, DOI 10.1017/CBO9780511530029
J. Solymosi and Cs. D. Tóth, Distinct distances in the plane, Discrete Comput. Geom. 25 (2001), no. 4, 629–634. The Micha Sharir birthday issue. MR 1838423, DOI 10.1007/s00454-001-0009-z
József Solymosi and Csaba D. Tóth, Distinct distances in homogeneous sets in Euclidean space, Discrete Comput. Geom. 35 (2006), no. 4, 537–549. MR 2225673, DOI 10.1007/s00454-006-1232-4
József Solymosi and Van Vu, Distinct distances in high dimensional homogeneous sets, Towards a theory of geometric graphs, Contemp. Math., vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp. 259–268. MR 2065269, DOI 10.1090/conm/342/06146
J. Spencer, E. Szemerédi, and W. Trotter Jr., Unit distances in the Euclidean plane, Graph theory and combinatorics (Cambridge, 1983) Academic Press, London, 1984, pp. 293–303. MR 777185
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
László A. Székely, Crossing numbers and hard Erdős problems in discrete geometry, Combin. Probab. Comput. 6 (1997), no. 3, 353–358. MR 1464571, DOI 10.1017/S0963548397002976
Endre Szemerédi and William T. Trotter Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), no. 3-4, 381–392. MR 729791, DOI 10.1007/BF02579194
Thomas Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996) Amer. Math. Soc., Providence, RI, 1999, pp. 129–162. MR 1660476
Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and a preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254, DOI 10.1090/ulect/029