Three-term polynomial progressions in subsets of finite fields

Sarah Peluse

doi:10.1007/s11856-018-1768-z

Three-term polynomial progressions in subsets of finite fields

Sarah Peluse

Mathematics

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

Abstract

Bourgain and Chang recently showed that any subset of F_p of density ≫p^−1/15 contains a nontrivial progression x, x + y, x + y². We answer a question of theirs by proving that if P₁, P₂ ∈ ℤ[y] are linearly independent and satisfy P₁(0) = P₂(0) = 0, then any subset of F_p of density ≫P₁,P₂p^−1/24 contains a nontrivial polynomial progression x, x + P₁(y), x + P₂(y).

Original language	English (US)
Pages (from-to)	379-405
Number of pages	27
Journal	Israel Journal of Mathematics
Volume	228
Issue number	1
DOIs	https://doi.org/10.1007/s11856-018-1768-z
State	Published - Oct 1 2018

All Science Journal Classification (ASJC) codes

General Mathematics

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title = "Three-term polynomial progressions in subsets of finite fields",

abstract = "Bourgain and Chang recently showed that any subset of Fp of density ≫p−1/15 contains a nontrivial progression x, x + y, x + y2. We answer a question of theirs by proving that if P1, P2 ∈ ℤ[y] are linearly independent and satisfy P1(0) = P2(0) = 0, then any subset of Fp of density ≫P1,P2p−1/24 contains a nontrivial polynomial progression x, x + P1(y), x + P2(y).",

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AB - Bourgain and Chang recently showed that any subset of Fp of density ≫p−1/15 contains a nontrivial progression x, x + y, x + y2. We answer a question of theirs by proving that if P1, P2 ∈ ℤ[y] are linearly independent and satisfy P1(0) = P2(0) = 0, then any subset of Fp of density ≫P1,P2p−1/24 contains a nontrivial polynomial progression x, x + P1(y), x + P2(y).

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Three-term polynomial progressions in subsets of finite fields

Abstract

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