Averages over hyperplanes, sumproduct theory in vector spaces over finite fields and the ErdősFalconer distance conjecture
Authors:
Derrick Hart, Alex Iosevich, Doowon Koh and Misha Rudnev
Journal:
Trans. Amer. Math. Soc. 363 (2011), 32553275
MSC (2010):
Primary 42B05, 11T23, 52C10
Published electronically:
December 29, 2010
MathSciNet review:
2775806
Fulltext 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 ErdősFalconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean ErdősFalconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the ErdősFalconer distance problem for subsets of the unit sphere in and discuss their sharpness. This results in a reasonably complete description of the ErdősFalconer distance problem in higherdimensional vector spaces over general finite fields.
 1.
Noga
Alon and Michael
Krivelevich, Constructive bounds for a Ramseytype problem,
Graphs Combin. 13 (1997), no. 3, 217–225. MR 1469821
(98h:05136), http://dx.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
(2007e:11092), http://dx.doi.org/10.1112/S0024610706022721
 3.
J.
Bourgain, N.
Katz, and T.
Tao, A sumproduct estimate in finite fields, and
applications, Geom. Funct. Anal. 14 (2004),
no. 1, 27–57. MR 2053599
(2005d:11028), http://dx.doi.org/10.1007/s0003900404511
 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
(2005i:11028)
 5.
M.
Burak Erdogan, A bilinear Fourier extension theorem and
applications to the distance set problem, Int. Math. Res. Not.
23 (2005), 1411–1425. MR 2152236
(2006h:42020), http://dx.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
(7,471c)
 7.
M.
Z. Garaev, The sumproduct estimate for large
subsets of prime fields, Proc. Amer. Math.
Soc. 136 (2008), no. 8, 2735–2739. MR 2399035
(2009e:11043), http://dx.doi.org/10.1090/S0002993908093866
 8.
A.
A. Glibichuk, Combinatorial properties of sets of residues modulo a
prime and the ErdősGraham problem, Mat. Zametki
79 (2006), no. 3, 384–395 (Russian, with
Russian summary); English transl., Math. Notes 79 (2006),
no. 34, 356–365. MR 2251362
(2007e:11120), http://dx.doi.org/10.1007/s1100600600408
 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
(2009a:11054)
 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
(2009m:11032), http://dx.doi.org/10.1090/conm/464/09080
 11.
D. Hart, A. Iosevich and J. Solymosi. Sumproduct 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 (electronic). MR 2336319
(2008k:11130), http://dx.doi.org/10.1090/S0002994707042651
 13.
A. Iosevich, M. Rudnev and I. UriarteTuero. Theory of dimension for large discrete sets and applications. Preprint, arxiv.org, 2007.
 14.
Nets
Hawk Katz and ChunYen
Shen, Garaev’s inequality in finite fields not of prime
order, Online J. Anal. Comb. 3 (2008), Art. 3, 6. MR 2375606
(2008k:12004)
 15.
Rudolf
Lidl and Harald
Niederreiter, Finite fields, Encyclopedia of Mathematics and
its Applications, vol. 20, AddisonWesley Publishing Company, Advanced
Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR 746963
(86c:11106)
 16.
Jiří
Matoušek, Lectures on discrete geometry, Graduate Texts
in Mathematics, vol. 212, SpringerVerlag, New York, 2002. MR 1899299
(2003f:52011)
 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
(90a:42009), http://dx.doi.org/10.1112/S0025579300013462
 18.
Elias
M. Stein, Harmonic analysis: realvariable 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
(95c:42002)
 19.
Terence
Tao and Van
Vu, Additive combinatorics, Cambridge Studies in Advanced
Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR 2289012
(2008a:11002)
 20.
Van
H. Vu, Sumproduct estimates via directed expanders, Math.
Res. Lett. 15 (2008), no. 2, 375–388. MR 2385648
(2009e:11023), http://dx.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
(10,234e)
 1.
 N. Alon and M. Krivelevich, Constructive bounds for a Ramseytype problem, Graphs and Combinatorics 13 (1997), 217225. MR 1469821 (98h:05136)
 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), 380398. MR 2225493 (2007e:11092)
 3.
 J. Bourgain, N. Katz and T. Tao. A sumproduct estimate in finite fields, and applications. Geom. Funct. Anal. 14 (2004), 2757. MR 2053599 (2005d:11028)
 4.
 E. Croot. Sums of the Form modulo a prime. Integers 4 (2004). MR 2116005 (2005i:11028)
 5.
 B. Erdogan. A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Not. 23 (2005), 14111425. MR 2152236 (2006h:42020)
 6.
 P. Erdős. On sets of distances of n points. Amer. Math. Monthly 53 (1946), 248250. MR 0015796 (7:471c)
 7.
 M. Garaev. The sumproduct estimate for large subsets of prime fields, Proc. Amer. Math. Soc. 136 (2008), 27352739. MR 2399035 (2009e:11043)
 8.
 A. A. Glibichuk. Combinatorial properties of sets of residues modulo a prime and the ErdősGraham problem. Mat. Zametki 79 (2006), 384395; translation in: Math. Notes 79 (2006), 356365. MR 2251362 (2007e:11120)
 9.
 A. Glibichuk and S. Konyagin. Additive properties of product sets in fields of prime order. Centre de Recherches Mathématiques, Proceedings and Lecture Notes, 2006. MR 2359478 (2009a:11054)
 10.
 D. Hart and A. Iosevich. Sums and products in finite fields: an integral geometric viewpoint, Contemporary Mathematics: Radon transforms, geometry, and wavelets 464 (2008). MR 2440133 (2009m:11032)
 11.
 D. Hart, A. Iosevich and J. Solymosi. Sumproduct 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), 61276142. MR 2336319 (2008k:11130)
 13.
 A. Iosevich, M. Rudnev and I. UriarteTuero. Theory of dimension for large discrete sets and applications. Preprint, arxiv.org, 2007.
 14.
 N. H. Katz and C.Y. Shen. Garaev's Inequality in finite fields not of prime order. Online J. Anal. Comb. No. 3 (2008), Art. 3, 6 pp. MR 2375606 (2008k:12004)
 15.
 R. Lidl and H. Niederrieter. Finite Fields. Encyclopedia of Mathematics and its Applications 20, AddisonWesley 1983. MR 746963 (86c:11106)
 16.
 J. Matousek. Lectures on Discrete Geometry, Graduate Texts in Mathematics. Springer 202, 2002. MR 1899299 (2003f:52011)
 17.
 P. Mattila. Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets. Mathematika 34 (2) (1987), 207228. MR 933500 (90a:42009)
 18.
 E. Stein. Harmonic Analysis. Princeton University Press, 1993. MR 1232192 (95c:42002)
 19.
 T. Tao and V. Vu. Additive Combinatorics. Cambridge University Press, 2006. MR 2289012 (2008a:11002)
 20.
 V. Vu. Sumproduct estimates via directed expanders. Math. Res. Lett. 15 (2008), no. 2, 375388. MR 2385648 (2009e:11023)
 21.
 A. Weil. On some exponential sums. Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204207. MR 0027006 (10:234e)
Similar Articles
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 088548019
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:
http://dx.doi.org/10.1090/S000299472010052328
PII:
S 00029947(2010)052328
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.
