A slight improvement to Garaev’s sum product estimate

By Nets Hawk Katz, Chun-Yen Shen. 2007

85 citations recorded by the source corpus.

max(\A + A\,\AA\)>Ce\A\?-*. In the finite field setting this situation is much more complicated because the main tool, the Szemer?di-Trotter incidence theorem, does not hold in the same generality. It is known, via the work in [BKT], that if A is a subset of Fp, the field of p elements with p prime, and if p6 0, then one has the sum product estimate max(|? + A|,|AA|)>|,4|1+ for some e > 0. This result has found many applications in combinatorial problems and exponential sum estimates (see e.g. [BKT], [BGK], [G2]). Recently, Garaev [Gl] showed that when \A\ < pz, one has the estimate

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/60c6cce29db41747b15b40c01ec6b7cb897ce16e

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In Artificial Intelligence

1. M. Rudnev
2. Moubariz Z. Garaev
3. Harald Andrés Helfgott
4. Chun-Yen Shen

In Discrete Mathematics and Combinatorics

1. M. Rudnev
2. Harald Andrés Helfgott
3. Chun-Yen Shen
4. Moubariz Z. Garaev
5. Ilya D. Shkredov
6. Sophie Stevens
7. Frank de Zeeuw
8. P. P. Varj'u

Citations in the corpus

• On the size of the set A(A +  1) (2008, methodology): “Using the step of Katz and Shen [11], we deduce that in either case there exist elements b1, b2, b3, b4 ∈ A1 such that”
• THE SUM-PRODUCT ESTIMATE FOR LARGE SUBSETS OF PRIME FIELDS (2007, unknown): “≫ |A|14/13(log |A|)O(1).”
• An explicit incidence theorem in p (2010, methodology): “Preprint arXiv:math/0304217 (2003), 9pp. [8] N.H. Katz, C.-Y. Shen.”
• An improved sum-product inequality in fields of prime order (2010, background): “The first one is a slight, but very useful generalisation of the Plünnecke-Ruzsa inequality (see [8]), which is due to Katz and Shen ([6]).”