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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 沈俊嚴(Chun-Yen Shen) | |
| dc.contributor.author | Guo-Dong Hong | en |
| dc.contributor.author | 洪國棟 | zh_TW |
| dc.date.accessioned | 2022-11-23T09:32:12Z | - |
| dc.date.available | 2022-02-21 | |
| dc.date.available | 2022-11-23T09:32:12Z | - |
| dc.date.copyright | 2022-02-21 | |
| dc.date.issued | 2022 | |
| dc.date.submitted | 2022-01-22 | |
| dc.identifier.citation | [1] J. Bourgain, Hausdorff dimension and distance sets, Israel J. Math. 87, 193-201, 1994. [2] X. Du, R. Zhang, Sharp L2 estimates of the Schr ̈odinger maximal function in higher dimensions. Ann. of Math. (2), 189(3):837-861, 2019. [3] X. Du, L. Guth, Y. Ou, H. Wang, B. Wilson, R. Zhang, Weighted restriction estimates and application to falconer distance set problem. American Journal of Mathematics Vol. 143, pp. 175-211, 2021. [4] M. B. Erdo ̆gan, A note on the Fourier transform of fractal measures, Math. Res. Lett. 11, 299-313, 2004. [5] M. B. Erdo ̆gan, A bilinear Fourier extension problem and applications to the distance set problem, Int. Math. Res. Not. 23, 1411-1425, 2005. [6] M. B. Erdo ̆gan, On Falconer’s distance set conjecture, Rev. Mat. Iberoam. 22, 649-662, 2006. [7] K. J. Falconer, Hausdorff dimension and the exceptional set of projections, Mathematika, 29(1):109-115, 1982. [8] K. J. Falconer, On the Hausdorff dimension of distance sets, Mathematika 32, 206-212, 1985. [9] A. Greenleaf, A. Iosevich, B. Liu, and E. Palsson, A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral, Rev. Mat. Iberoam., 31(3):799-810, 2015. [10] L. Guth, A. Iosevich, Y. Ou, H. Wang, On falconer’s distance set problem in the plane. Inventiones mathematicae vol. 219, pages779-830, 2020. [11] V. Harangi, T. Keleti, G. Kiss, P. Maga, A. M ́ath ́e, P. Mattila, B. Strenner, How large dimension guarantees a given angle? Monatshefte f ̈ur Mathematik volume 171, 169-187, 2013. [12] A. Iosevich and E. Palsson, An improved dimensional threshold for the angle problem, arXiv:1807.05465 ,2019. [13] A. Iosevich, M. Mourgoglou and E. Palsson, On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators, Mathematical Research Letters, 23, 1737-1759, 2016. [14] B. Liu, Group actions, the Mattila integral and applications, Proc. Amer. Math. Soc., 2017. [15] B. Liu, An L2-identity and pinned distance problem, Geom. Funct. Anal., 29(1):283-294, 2019. [16] B. Liu, Hausdorff dimension of pinned distance sets and the L2-method, Proc. Amer. Math. Soc. 148, 333-341, 2020. [17] P. Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika 34, 207-228. 1987. [18] P. Mattila, Hausdorff dimension, projections, and the Fourier transform, Publ. Mat. 48(1), 3-48, 2004. [19] P. Mattila, Fourier analysis and Hausdorff dimension, Vol. 150, Cambridge University Press, 2015. [20] J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3), 4:257-302, 1954. [21] T. Orponen, lecture note of Geometric measure theory, 2018. [22] T. Orponen, On the dimension and smoothness of radial projections, Anal. PDE, 12(5):1273-1294, 2019. [23] Y. Peres and W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102, 193-251 ,2000. [24] E. Stein and R. Sharkarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis). [25] T. W. Wolff, Decay of circular means of Fourier transforms of measures, Int. Math. Res. Not. 10, 547-567, 1999. [26] T. W. Wolff, Lectures on Harmonic Analysis, Amer. Math. Soc., University Lecture Series 29. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/80214 | - |
| dc.description.abstract | 在現代的幾何測度論中Falconer距離問題是其中一個廣為人知的猜想。近年來許多新的手法陸續的被提出，而群作用方式就是其中一種。在這篇碩士論文當中，我們考慮了距離問題的變形版本:角度問題，並連結到徑向投影。特別的是我們用了群作用方式得到了兩個關於Salem 集合徑向投影的新結果。 | zh_TW |
| dc.description.provenance | Made available in DSpace on 2022-11-23T09:32:12Z (GMT). No. of bitstreams: 1 U0001-2101202214441000.pdf: 682072 bytes, checksum: 794c9f3760589ebcb680aa10f179c177 (MD5) Previous issue date: 2022 | en |
| dc.description.tableofcontents | Introduction 1 1.1 Projection problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Hausdorff dimension 4 2.1 Hausdorff measure and Hausdorff dimension . . . . . . . . . . . . . . 4 2.2 Frostman’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Riesz energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Fourier transform 9 3.1 Fourier transform of measure . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Energy integral via Fourier transform . . . . . . . . . . . . . . . . . . 10 3.3 L2-spherical averages . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Salem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Falconer’s distance problem 13 4.1 Steinhaus’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Classical results and conjecture . . . . . . . . . . . . . . . . . . . . . 14 4.3 Mattila’s paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.4 The decay of L2-spherical averages . . . . . . . . . . . . . . . . . . . 17 4.5 Group action method . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.6 Pinned distance problem . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.7 Remark on recent results . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Angle problem and radial projection 27 5.1 Angle problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Radial projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3 Orponen’s result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 New results: radial projection and the group action method 35 6.1 Radial projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Pinned radial projection . . . . . . . . . . . . . . . . . . . . . . . . . 38 References 45 | |
| dc.language.iso | en | |
| dc.subject | Salem 集合 | zh_TW |
| dc.subject | 群作用 | zh_TW |
| dc.subject | Falconer 距離問題 | zh_TW |
| dc.subject | 徑向投影 | zh_TW |
| dc.subject | 角度問題 | zh_TW |
| dc.subject | Hausdorff 維數 | zh_TW |
| dc.subject | Hausdorff dimension | en |
| dc.subject | Salem set | en |
| dc.subject | radial projection | en |
| dc.subject | group action | en |
| dc.subject | Falconer’s distance problem | en |
| dc.subject | angle problem | en |
| dc.title | 投影問題與群作用方法 | zh_TW |
| dc.title | Projection Problems and The Group Action Method | en |
| dc.date.schoolyear | 110-1 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 夏俊雄(Emmy Chang),陳逸昆,司靈得 | |
| dc.subject.keyword | 角度問題,Falconer 距離問題,群作用,Hausdorff 維數,徑向投影,Salem 集合, | zh_TW |
| dc.subject.keyword | angle problem,Falconer’s distance problem,group action,Hausdorff dimension,radial projection,Salem set, | en |
| dc.relation.page | 47 | |
| dc.identifier.doi | 10.6342/NTU202200141 | |
| dc.rights.note | 同意授權(全球公開) | |
| dc.date.accepted | 2022-01-22 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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