Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-01T20:05:40.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields

Published online by Cambridge University Press:  20 November 2018

Doowon Koh  and
Chun-Yen Shen
Doowon Koh
Affiliation:
Department of Mathematics, Chungbuk National University, Cheongju city, Chungbuk-Do 361-736, Korea email: koh131@chungbuk.ac.kr
Chun-Yen Shen
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1 email: shenc@umail.iu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study the extension problem, the averaging problem, and the generalized Erdős–Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Keywords

42B0511T2452C17extension problemsaveraging operatorfinite fieldsErdős–Falconer distance problemshomogeneous polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 5 , 01 October 2012 , pp. 1036 - 1057
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Bourgain, J., Katz, N., and Tao, T., A sum-product estimate in finite fields, and applications. Geom. Funct. Anal. 14(2004), no. 1, 27–57. http://dx.doi.org/10.1007/s00039-004-0451-1 Google Scholar
[2] Carbery, A., Stones, B., and J.Wright, Averages in vector spaces over finite fields. Math. Proc. Cambridge Philos. Soc. 144(2008), no. 1, 13–27.Google Scholar
[3] Chapman, J., Erdo˜gan, M., Hart, D., Iosevich, A., and Koh, D., Pinned distance sets, Wolff 's exponent in finite fields and sum-product estimates. Math. published online, Z., March 22, 2011. http://dx.doi.org/10.1007/s00209-011-0852-4 Google Scholar
[4] Cochrane, T., Exponential sums and the distribution of solutions of congruences, Inst. of Math., Academia Sinica, Taipei, 1994.Google Scholar
[5] Erdo˜gan, M., A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Not. 23(2005), 1411–1425.Google Scholar
[6] Erdʺos, P., On sets of distances of n points. Amer. Math. Monthly 53(1946), 248–250. http://dx.doi.org/10.2307/2305092 Google Scholar
[7] Falconer, K. J., On the Hausdorff dimensions of distance sets. Mathematika 32(1985), no. 2, 206–212. http://dx.doi.org/10.1112/S0025579300010998 Google Scholar
[8] Guth, L. and Katz, N., On the Erdʺos distinct distance problem in the plane. Ann of Math., to appear.Google Scholar
[9] Hart, D., Iosevich, A., Koh, D., and Rudnev, M., Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdös-Falconer distance conjecture. Trans. Amer. Math. Soc. 363(2011), no.6, 3255–3275. http://dx.doi.org/10.1090/S0002-9947-2010-05232-8 Google Scholar
[10] Iosevich, A. and Koh, D., Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields. Illinois J. Math. 52(2008), no. 2, 611–628.Google Scholar
[11] Iosevich, A. and Rudnev, M., Erdös distance problem in vector spaces over finite fields. Trans. Amer. Math. Soc. 359(2007), no. 12, 6127–6142. http://dx.doi.org/10.1090/S0002-9947-07-04265-1 Google Scholar
[12] Iosevich, A. and Sawyer, E., Sharp Lp − Lq estimates for a class of averaging operators. Ann. Inst. Fourier (Grenoble) 46(1996), no. 5, 1359–1384. http://dx.doi.org/10.5802/aif.1553 Google Scholar
[13] Katz, N., Estimates for “singular” exponential sums. Internat. Math. Res. Notices 1999, no. 16, 875–899.Google Scholar
[14] Koh, D., Extension and averaging operators for finite fields. arxiv:0908.3266.Google Scholar
[15] Koh, D. and Shen, C., Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields. Revista Matematica Iberoamericana, to appear.Google Scholar
[16] Koh, D. and Shen, C., The generalized Erdös-Falconer distance problems in vector spaces over finite fields. arxiv:1004.4012.Google Scholar
[17] Littman, W., Lp − Lq estimates for singular integral operators from hyperbolic equations. In: Partial differential equations (Proc. Sympos. Pure Math., 23, Univ. California, Berkeley, Calif., 1971), American Mathematical Society, Providence, RI, 1973, pp. 479–481.Google Scholar
[18] Lidl, R. and Niederreiter, H., Finite fields. Encyclopedia of Mathematics and it Applications, 20, Cambridge University Press, Cambridge, 1997.Google Scholar
[19] Mockenhaupt, G. and Tao, T., Restriction and Kakeya phenomena for finite fields. Duke Math. J. 121(2004), no. 1, 35–74. http://dx.doi.org/10.1215/S0012-7094-04-12112-8 Google Scholar
[20] Salié, H., Über die Kloostermanschen Summen S(u, v; q). Math. Z. 34(1932), no. 1, 91–109. http://dx.doi.org/10.1007/BF01180579 Google Scholar
[21] Stein, E. M., Lp boundedness of certain convolution operators. Bull. Amer. Math. Soc. 77(1971), 404–405. http://dx.doi.org/10.1090/S0002-9904-1971-12716-7 Google Scholar
[22] Stein, E. M., Some problems in harmonic analysis. In: Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Proc. Sympos. Pure Math., 35, American Mathematical Society, Providence, RI, 1979, pp. 3–20.Google Scholar
[23] Strichartz, R. S., Convolutions with kernels having singularities on the sphere. Trans. Amer. Math. Soc. 148(1970), 461–471. http://dx.doi.org/10.1090/S0002-9947-1970-0256219-1 Google Scholar
[24] Solymosi, J. and Tóth, C. D., Distinct distances in the plane. Discrete Comput. Geom. 25(2001), no. 4, 629–634.Google Scholar
[25] Solymosi, J. and Vu, V., Distinct distances in high dimensional homogeneous sets. In: Towards a theory of geometric graphs, Contemporary Mathematics, 342, American Mathematical Society, Providence, RI, 2004, pp. 259–268.Google Scholar
[26] Solymosi, J. and Vu, V., Near optimal bounds for the number of distinct distances in high dimensions. Combinatorica 28(2008), no. 1, 113–125. http://dx.doi.org/10.1007/s00493-008-2099-1 Google Scholar
[27] Tao, T., A sharp bilinear restriction estimate for paraboloids. Geom. Funct. Anal. 13(2003), no. 6, 1359–1384. http://dx.doi.org/10.1007/s00039-003-0449-0 Google Scholar
[28] Tao, T., Some recent progress on the restriction conjecture. In: Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 217–243.Google Scholar
[29] Weil, A., On some exponential sums. Proc. Nat. Acad. Sci. U. S. A. 34(1948), 204–207. http://dx.doi.org/10.1073/pnas.34.5.204 Google Scholar
[30] Wolff, T., Decay of circular means of Fourier transforms of measures. Internat. Math. Res. Notices 1999, no. 10, 547–567.Google Scholar