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Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the p-adic numbers

Part of: Diophantine approximation, transcendental number theory Harmonic analysis in several variables Abstract harmonic analysis Classical measure theory

Published online by Cambridge University Press:  29 January 2019

Robert Fraser  and
Kyle Hambrook
Robert Fraser
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T1Z2, Canada ([email protected])
Kyle Hambrook
Affiliation:
Department of Mathematics and Statistics, San Jose State University, San Jose, CA95192, USA ([email protected])
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Abstract

We exhibit the first explicit examples of Salem sets in ℚp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.

Keywords

Hausdorff dimensionFourier dimensionSalem setsFourier restrictionMetric Diophantine approximationp-adic

MSC classification

Primary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Secondary: 11J83: Metric theory 11J61: Approximation in non-Archimedean valuations 28A78: Hausdorff and packing measures 28A80: Fractals 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1265 - 1288
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Abercrombie, A. G.. The Hausdorff dimension of some exceptional sets of p-adic integer matrices. J. Number Theory, 53 (1995), 311–341.CrossRefGoogle Scholar
2Allen, D. and Troscheit, S.. The Mass Transference Principle: Ten Years On. https://arxiv.org/abs/1704.06628v2.Google Scholar
3Bak, J.-G. and Seeger, A.. Extensions of the Stein-Tomas theorem. Math. Res. Lett., 18 (2011), 767–781.CrossRefGoogle Scholar
4Beresnevich, V., Dickinson, D. and Velani, S.. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc., 179 (2006)Google Scholar
5Beresnevich, V., Bernik, V., Dodson, M. and Velani, S.. Classical metric Diophantine approximation revisited. In Analytic number theory, pages 38–61 (Cambridge: Univ Cambridge Press 2009).Google Scholar
6Besicovitch, A. S.. Sets of Fractional Dimensions (IV): On Rational Approximation to Real Numbers. J. London Math. Soc., S1–9 126, 1934.CrossRefGoogle Scholar
7Bluhm, C.. Random recursive construction of Salem sets. Ark. Mat., 34 (1996), 51–63.10.1007/BF02559506CrossRefGoogle Scholar
8Bluhm, C.. On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets. Ark. Mat., 36 (1998), 307–316.10.1007/BF02384771CrossRefGoogle Scholar
9Bogachev, V. I.. Measure theory. Vol. I, II (Berlin: Springer-Verlag, 2007).CrossRefGoogle Scholar
10Bovey, J. D. and Dodson, M. M.. The Hausdorff dimension of systems of linear forms. Acta Arith., 45 (1986), 337–358.CrossRefGoogle Scholar
11Carbery, A., Seeger, A., Wainger, S. and Wright, J.. Classes of singular integral operators along variable lines. J. Geom. Anal., 9 (1999), 583–605.CrossRefGoogle Scholar
12Chen, X. and Seeger, A.. Convolution powers of Salem measures with applications. Canad. J. Math., 69 (2017), 284–320.CrossRefGoogle Scholar
13Cilleruelo, J. and Garaev, M. Z.. Concentration of points on two and three dimensional modular hyperbolas and applications. Geom. Funct. Anal., 21 (2011), 892–904.CrossRefGoogle Scholar
14Eggleston, H. G.. Sets of fractional dimensions which occur in some problems of number theory. Proc. London Math. Soc. (2) 54 (1952), 42–93.CrossRefGoogle Scholar
15Ekström, F.. Fourier dimension of random images. Ark. Mat., 54 (2016), 455–471.CrossRefGoogle Scholar
16Ekström, F., Persson, T. and Schmeling, J.. On the Fourier dimension and a modification. J. Fractal Geom., 2 (2015), 309–337.CrossRefGoogle Scholar
17Falconer, K. J.. The geometry of fractal sets, volume 85 Cambridge Tracts in Mathematics (Cambridge: Cambridge University Press, 1986).Google Scholar
18Folland, G. B.. A course in abstract harmonic analysis. Studies in Advanced Mathematics. (Boca Raton FL: CRC Press, 1995).Google Scholar
19Fraser, J. M., Orponen, T. and Sahlsten, T.. On Fourier analytic properties of graphs. Int. Math. Res. Not. IMRN (2014), 2730–2745.CrossRefGoogle Scholar
20Hambrook, K.. Explicit Salem sets and applications to metrical Diophantine approximation. https://arxiv.org/abs/1604.00411v1.Google Scholar
21Hambrook, K.. Explicit Salem sets in ℝ2. Adv. Math. 311 (2017), 634–648.CrossRefGoogle Scholar
22Jarník, V.. Diophantischen Approximationen und Hausdorffsches Mass. Mat. Sborjnik 36 (1929), 371–382.Google Scholar
23Jarník, V.. über die simultanen diophantischen Approximationen. Math. Z., 33 (1931), 505–543.CrossRefGoogle Scholar
24Kahane, J.-P.. Some random series of functions, 2nd edn, volume. 5, of Cambridge Studies in Advanced Mathematics (Cambridge: Cambridge University Press, 1985)Google Scholar
25Kaufman, R.. On the theorem of Jarník and Besicovitch. Acta Arith., 39 (1981), 265–267.CrossRefGoogle Scholar
26Łaba, I. and Pramanik, M.. Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal., 19 (2009), 429–456.10.1007/s00039-009-0003-9CrossRefGoogle Scholar
27Mattila, P.. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics (Cambridge: Cambridge University Press, 1995). Fractals and rectifiability.CrossRefGoogle Scholar
28Mattila, P.. Fourier analysis and Hausdorff dimension, volume 150 of Cambridge Studies in Advanced Mathematics (Cambridge: Cambridge University Press, 2015).10.1017/CBO9781316227619CrossRefGoogle Scholar
29Melničuk, J. V.. Hausdorff dimension in Diophantine approximations of p-adic numbers, Ukrain. Mat. Zh., 32 (1980), 118–124.Google Scholar
30Mitsis, T.. A Stein-Tomas restriction theorem for general measures. Publ. Math. Debrecen 60 (2002), 89–99.Google Scholar
31Mockenhaupt, G.. Bounds in Lebesgue spaces of oscillatory integral operators (Siegen: Habilitationsschrift, Gesamthochschule, 1996).Google Scholar
32Mockenhaupt, G.. Salem sets and restriction properties of Fourier transforms. Geom. Funct. Anal. 10 (2000), 1579–1587.CrossRefGoogle Scholar
33Mockenhaupt, G. and Ricker, W. J.. Optimal extension of the Hausdorff-Young inequality. J. Reine Angew. Math. 620 (2008), 195–211.Google Scholar
34Papadimitropoulos, C.. The Fourier restriction phenomenon in thin sets. PhD thesis, University of Edinburgh, (2010a).Google Scholar
35Papadimitropoulos, C.. Salem sets in local fields, the Fourier restriction phenomenon and the Hausdorff-Young inequality. J. Funct. Anal. 259 (2010b), 1–27.CrossRefGoogle Scholar
36Papadimitropoulos, C.. Salem sets in the p-adics, the Fourier restriction phenomenon and optimal extension of the Hausdorff-Young inequality. In (Curbera, Guillermo P., Mockenhaupt, Gerd and Ricker, Werner J., eds) Vector measures, integration and related topics, volume 201 of Oper Theory Adv. Appl. pages 327–338 (Basel: Birkhäuser Verlag 2010c).Google Scholar
37Salem, R.. On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1 (1951), 353–365.CrossRefGoogle Scholar
38Shmerkin, P. and Suomala, V.. Spatially independent martingales, intersections, and applications. To appear in Memoirs of the Amer. Math. Soc. https://arxiv.org/abs/1409.6707v4.Google Scholar
39Stein, E. M.. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series (Princeton, NJ: Princeton University Press 1993.) With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.Google Scholar
40Taibleson, M. H.. Fourier analysis on local fields (Princeton, NJ, Tokyo: Princeton University Press, University of Tokyo Press, 1975).Google Scholar