Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture

Authors:
Derrick Hart, Alex Iosevich, Doowon Koh and Misha Rudnev

Journal:
Trans. Amer. Math. Soc. **363** (2011), 3255-3275

MSC (2010):
Primary 42B05, 11T23, 52C10

DOI:
https://doi.org/10.1090/S0002-9947-2010-05232-8

Published electronically:
December 29, 2010

MathSciNet review:
2775806

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering , the finite field with elements, by , where is a subset of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdős-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdős-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdős-Falconer distance problem for subsets of the unit sphere in and discuss their sharpness. This results in a reasonably complete description of the Erdős-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

**1.**Noga Alon and Michael Krivelevich,*Constructive bounds for a Ramsey-type problem*, Graphs Combin.**13**(1997), no. 3, 217–225. MR**1469821**, https://doi.org/10.1007/BF03352998**2.**J. Bourgain, A. A. Glibichuk, and S. V. Konyagin,*Estimates for the number of sums and products and for exponential sums in fields of prime order*, J. London Math. Soc. (2)**73**(2006), no. 2, 380–398. MR**2225493**, https://doi.org/10.1112/S0024610706022721**3.**J. Bourgain, N. Katz, and T. Tao,*A sum-product estimate in finite fields, and applications*, Geom. Funct. Anal.**14**(2004), no. 1, 27–57. MR**2053599**, https://doi.org/10.1007/s00039-004-0451-1**4.**Ernie Croot,*Sums of the form 1/𝑥^{𝑘}₁+…+1/𝑥^{𝑘}_{𝑛} modulo a prime*, Integers**4**(2004), A20, 6. MR**2116005****5.**M. Burak Erdo an,*A bilinear Fourier extension theorem and applications to the distance set problem*, Int. Math. Res. Not.**23**(2005), 1411–1425. MR**2152236**, https://doi.org/10.1155/IMRN.2005.1411**6.**P. Erdös,*On sets of distances of 𝑛 points*, Amer. Math. Monthly**53**(1946), 248–250. MR**0015796**, https://doi.org/10.2307/2305092**7.**M. Z. Garaev,*The sum-product estimate for large subsets of prime fields*, Proc. Amer. Math. Soc.**136**(2008), no. 8, 2735–2739. MR**2399035**, https://doi.org/10.1090/S0002-9939-08-09386-6**8.**A. A. Glibichuk,*Combinatorial properties of sets of residues modulo a prime and the Erdős-Graham problem*, Mat. Zametki**79**(2006), no. 3, 384–395 (Russian, with Russian summary); English transl., Math. Notes**79**(2006), no. 3-4, 356–365. MR**2251362**, https://doi.org/10.1007/s11006-006-0040-8**9.**A. A. Glibichuk and S. V. Konyagin,*Additive properties of product sets in fields of prime order*, Additive combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 279–286. MR**2359478****10.**Derrick Hart and Alex Iosevich,*Sums and products in finite fields: an integral geometric viewpoint*, Radon transforms, geometry, and wavelets, Contemp. Math., vol. 464, Amer. Math. Soc., Providence, RI, 2008, pp. 129–135. MR**2440133**, https://doi.org/10.1090/conm/464/09080**11.**D. Hart, A. Iosevich and J. Solymosi.*Sum-product theorems in finite fields via Kloosterman sums*. Int. Math. Res. Notices (2007) Vol. 2007, article ID rmn007, 14 pages.**12.**A. Iosevich and M. Rudnev,*Erdős distance problem in vector spaces over finite fields*, Trans. Amer. Math. Soc.**359**(2007), no. 12, 6127–6142. MR**2336319**, https://doi.org/10.1090/S0002-9947-07-04265-1**13.**A. Iosevich, M. Rudnev and I. Uriarte-Tuero.*Theory of dimension for large discrete sets and applications.*Preprint, arxiv.org, 2007.**14.**Nets Hawk Katz and Chun-Yen Shen,*Garaev’s inequality in finite fields not of prime order*, Online J. Anal. Comb.**3**(2008), Art. 3, 6. MR**2375606****15.**Rudolf Lidl and Harald Niederreiter,*Finite fields*, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR**746963****16.**Jiří Matoušek,*Lectures on discrete geometry*, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR**1899299****17.**Pertti Mattila,*Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets*, Mathematika**34**(1987), no. 2, 207–228. MR**933500**, https://doi.org/10.1112/S0025579300013462**18.**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****19.**Terence Tao and Van Vu,*Additive combinatorics*, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR**2289012****20.**Van H. Vu,*Sum-product estimates via directed expanders*, Math. Res. Lett.**15**(2008), no. 2, 375–388. MR**2385648**, https://doi.org/10.4310/MRL.2008.v15.n2.a14**21.**André Weil,*On some exponential sums*, Proc. Nat. Acad. Sci. U. S. A.**34**(1948), 204–207. MR**0027006**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
42B05,
11T23,
52C10

Retrieve articles in all journals with MSC (2010): 42B05, 11T23, 52C10

Additional Information

**Derrick Hart**

Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

**Alex Iosevich**

Affiliation:
Department of Mathematics, University of Rochester, Hylan 909, Rochester, New York 14627

**Doowon Koh**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

**Misha Rudnev**

Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW England

DOI:
https://doi.org/10.1090/S0002-9947-2010-05232-8

Received by editor(s):
May 28, 2008

Received by editor(s) in revised form:
September 26, 2009

Published electronically:
December 29, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.