A new sum-product estimate in prime fields
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TL;DR: In this paper, the authors obtained a new sum-product estimate in prime fields for sets of large cardinality using a point-plane incidence bound rather than the point-line incidence bound used by Shakan and Shkredov.
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Abstract: We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’, Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.
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Citations
Polynomial Methods and Incidence Theory
Adam Sheffer
- 17 Mar 2022
TL;DR: A detailed introduction to polynomial methods and their applications with a focus on incidence theory is given in this paper , with a minimal background, allowing graduate and advanced undergraduate students to get to grips with an active and exciting research front.
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Stronger sum--product inequalities for small sets
Misha Rudnev,Ilya D. Shkredov,George Shakan +2 more
- 06 Jan 2020
TL;DR: In this article, the threshold-breaking sum-product inequality was shown to hold regardless of the characteristic of a finite subset of a field and a finite subset of the field.
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•Posted Content
Stronger sum-product inequalities for small sets
TL;DR: In this paper, the threshold-breaking sum-product inequality was shown to hold regardless of the characteristic of a finite subset of a field and a finite subset of the field.
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On Products of Shifts in Arbitrary Fields
TL;DR: In this article, the authors adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field, for all $A \subset \mathbb{F}$ finite with $|A| < p^{1/4}$ if $p:= Char(\mathbb {F})$ is positive, they have
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Attaining the exponent $5/4$ for the sum-product problem in finite fields
Ali Mohammadi,Sophie Stevens +1 more
TL;DR: In this article, the authors improved the exponent in the finite field sum-product problem from $11/9$ to $5/4, improving the results of Rudnev, Shakan and Shkredov.
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References
A sum-product estimate in finite fields, and applications
TL;DR: In this paper, the Szemeredi-Trotter type theorem for finite fields was shown to be equivalent to the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem.
•Posted Content
A sum-product estimate in finite fields, and applications
TL;DR: A Szemerédi-Trotter type theorem in finite fields is proved, and a new estimate for the Erdös distance problem in finite field, as well as the three-dimensional Kakeya problem in infinite fields is obtained.
343
On sums and products of integers
P. Erdős,Endre Szemerédi +1 more
- 01 Jan 1983
TL;DR: In this paper, the integers of the form ======¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯�ﾷ (1) were considered, where n is a sequence of integers, and n is the number of nodes in the sequence.
288
On the number of sums and products
TL;DR: In this article, the authors improved the exponent to 5/4, which is the smallest lower bound known for any n-element set A, and they showed that the lower bound can be obtained by a positive absolute constant c 1 such that g(n) ≥ n 1+c 1.
An improved point‐line incidence bound over arbitrary fields
Sophie Stevens,Frank de Zeeuw +1 more
TL;DR: In this paper, an upper bound for the number of incidences between points and lines in a plane over an arbitrary field F was shown, which was later improved to O(m 11/15n11/15) by using Cartesian products.
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