Sum-product Estimates in Finite Fields via Kloosterman Sums

Hart, Derrick; Iosevich, Alex; Solymosi, József

doi:10.1093/imrn/rnm007

Cited by 56 publications

(91 citation statements)

References 17 publications

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“…This enabled the first two authors and J. Solymosi to use its analog to obtain a nontrivial sum-product estimate ( [11]). The basis for the latter results was Weil's bound ( [21]) for Kloosterman sums,…”

Section: Theorem 13 ([12]) Suppose Thatmentioning

confidence: 99%

“…[9]) that if d = 3 and q is prime, this conclusion holds if the number of elements |A| ≥ Cq fact be a subfield. See also [2], [4], [8], [7], [11], [14], [19], [20] and the references contained therein on recent progress related to this problem and its analogs. For example, Glibichuk, [8], proved that 8A · B = Z p , p prime, provided that |A||B| > p and either A = −A or A ∩ (−A) = ∅.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Hart

Iosevich

Koh

et al. 2010

Trans. Amer. Math. Soc.

142

146

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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 F q , the finite field with q elements, by A · A + · · · + A · A, where A is a subset F q 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 F d q 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.

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Section: Theorem 13 ([12]) Suppose Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Hart

Iosevich

Koh

et al. 2010

Trans. Amer. Math. Soc.

142

146

View full text Add to dashboard Cite

show abstract

“…Clearly (5) is not implied by (10) and (11). On the other hand, we show that our approach also gives an alternative proof of the corresponding results of [9] and [18] for available at https://www.cambridge.org/core/terms.…”

Section: Possible Improvementsmentioning

confidence: 54%

“…, A n , B n ; λ) to denote its number of solutions. In the case A 1 = B 1 = · · · = A n = B n = A, it is shown in [11], using bounds of Kloosterman sums, that (15) has a solution for any λ ∈ ‫ކ‬ q , provided that #A ≥ q 1/2+1/(2n) . Clearly, using multiplicative character sums one can also obtain new results for equation (15) (for example, in the case when the sets B 1 , .…”

Section: Further Problemsmentioning

confidence: 99%

On the Solvability of Bilinear Equations in Finite Fields

Shparlinski¹

2008

Glasgow Math. J.

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Abstract. We consider the equationover a finite field ‫ކ‬ q of q elements, with variables from arbitrary sets A, B, C, D ⊆ ‫ކ‬ q . The question of solvability of such and more general equations has recently been considered by Hart and Iosevich, who, in particular, prove that iffor some absolute constant C > 0, then above equation has a solution for any λ ∈ ‫ކ‬ * q . Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.2000 Mathematics Subject Classification. 11L40, 11T30.

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“…These kinds of results have found many important applications in various areas of mathematics, and people want to know some quantitative relationships between δ and ǫ in certain ranges of |A|. For the case |A| p 0.5 , the pioneer work was due to Hart, Iosevich and Solymosi [12] via Kloosterman sums. See [6,8,18] for further improvements.…”

Section: Introductionmentioning

confidence: 99%

Slightly improved sum-product estimates in fields of prime order

2011

Acta Arith.

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Abstract. Let Fp be the field of residue classes modulo a prime number p and let A be a nonempty subset of Fp. In this paper we show that if |A| p 0.5 , then max{|A ± A|, |AA|} |A| 13/12 ;These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on different sets are also considered. IntroductionLet F p be the field of residue classes modulo a prime number p and let A, B be two nonempty subsets of F p . Define the sum set, difference set and product set of A and B respectively byFrom the work of Bourgain, Katz, Tao [4] and Bourgain, Glibichuk, Konyagin [3], it is known that if |A| p δ (see Section 2 for the definitions of , , , and ∼), where δ < 1, then one has the sum-product estimate max{|A + A|, |AA|} |A| 1+ǫ for some ǫ = ǫ(δ) > 0. These kinds of results have found many important applications in various areas of mathematics, and people want to know some quantitative relationships between δ and ǫ in certain ranges of |A|. For the case |A| p 0.5 , the pioneer work was due to Hart, Iosevich and Solymosi [12] via Kloosterman sums. See [6,8,18] for further improvements. Note also all lower bounds in [6,8,12,18]

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Sum-product Estimates in Finite Fields via Kloosterman Sums

Abstract: We establish improved sum-product bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.1991 Mathematics Subject Classification. 42B10;81S30;94A12.

Cited by 56 publications

References 17 publications

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

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

On the Solvability of Bilinear Equations in Finite Fields

Slightly improved sum-product estimates in fields of prime order

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