A Cauchy-Davenport Theorem for Linear Maps

Abstract

We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets A, B of the finite field \(\mathbb{F}_p \), the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset A + B in terms of the sizes of the sets A and B. Our theorem considers a general linear map \(L:\mathbb{F}_p^n \to \mathbb{F}_p^m \), and subsets \(A_1 , \ldots A_n \subseteq \mathbb{F}_p\), and gives a lower bound on the size of L(A1 × A2 × … × An) in terms of the sizes of the sets A1, …, An.

Our proof uses Alon’s Combinatorial Nullstellensatz and a variation of the polynomial method.

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Correspondence to John Kim.

Additional information

Research supported in part by NSF Grant Number DGE-1433187.

Research supported in part by a Sloan Fellowship and NSF grant CCF-1253886.

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Herdade, S., Kim, J. & Koppartyy, S. A Cauchy-Davenport Theorem for Linear Maps. Combinatorica 38, 601–618 (2018). https://doi.org/10.1007/s00493-016-3486-7

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Mathematics Subject Classification (2000)

  • 11B13
  • 15A04
  • 11T06
  • 05E15