An improved sum-product estimate for general finite fields

Li, Liangpan; Roche-Newton, Oliver

doi:10.1137/110823122

Cited by 21 publications

(28 citation statements)

References 10 publications

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“…For instance, we establish a new sum-product bound |A ± A| + |A · A| = Ω |A| 1+ 1 5 , for any A ⊂ F such that |A| < p 5 8 . This is a considerable improvement over the previously established best results in [25], [22], which were based on purely arithmetic techniques. In spirit, our main result is akin to the well-known sum-product estimate of Elekes, [9], yielding the exponent 1 + 1 4 for reals.…”

Section: Introductionmentioning

confidence: 64%

New sum-product type estimates over finite fields

Roche-Newton

Rudnev

Shkredov

2016

Advances in Mathematics

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Let F be a field with positive odd characteristic p. We prove a variety of new sum-product type estimates over F . They are derived from the theorem that the number of incidences between m points and n planes in the projective three-spacewhere k denotes the maximum number of collinear planes.The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely thatfor any A such that |A| < p 5 8 .for the sumset A + A of A ⊆ F , and similarly for the product set AA, alias A · A. Sometimes we write nA for multiple sumsets, e.g. A+A+A = 3A, as well as A −1 = {a −1 : a ∈ A\{0}}.

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Section: Introductionmentioning

confidence: 64%

New sum-product type estimates over finite fields

Roche-Newton

Rudnev

Shkredov

2016

Advances in Mathematics

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“…The following result is a small modification of [16, Theorem 1.4]. The necessary sum-ratio estimate is mentioned (without a proof) in [16]; a full proof can be found in [19]. Lemma 1.…”

Section: Polynomial Version Of Sum-product Estimatesmentioning

confidence: 99%

Polynomial Values in Subfields and Affine Subspaces of Finite Fields

Roche-Newton

Shparlinski

2014

The Quarterly Journal of Mathematics

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“…It should also be remarked that by combining recent sum-product work of Li and Roche-Newton [6] with the approach in [5], the exponent in Theorem 3 extends to a general finite field F q , subject to appropriate non-degeneracy conditions. This is omitted in the interests of simplicity and instead left as an exercise for the motivated reader.…”

Section: Resultsmentioning

confidence: 99%

An improved incidence bound for fields of prime order

Jones

2016

European Journal of Combinatorics

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We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field F, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in F 2 , with m 7/8 < n < m 8/7 , determine at most O(m 11/15 n 11/15) incidences (where, if F has positive characteristic p, we assume m −2 n 13 ≪ p 15). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.

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An improved sum-product estimate for general finite fields

Abstract: This paper improves on a sum-product estimate obtained by Katz and Shen for subsets of a finite field whose order is not prime.

Cited by 21 publications

References 10 publications

New sum-product type estimates over finite fields

New sum-product type estimates over finite fields

Polynomial Values in Subfields and Affine Subspaces of Finite Fields

An improved incidence bound for fields of prime order

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