Bounds for sets with no polynomial progressions

Sarah Peluse

doi:10.1017/fmp.2020.11

Bounds for sets with no polynomial progressions

Sarah Peluse

Mathematics

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

15 Scopus citations

Abstract

Let P₁, . . ., P_m ∈ ℤ[y] be polynomials with distinct degrees, each having zero constant term. We show that any subset A of {1, . . ., N} with no nontrivial progressions of the form x, x + P₁(y), . . ., x + P_m(y) has size |A| ≪ N/(log log N)^cP1^,...,Pm . Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

Original language	English (US)
Article number	e16
Journal	Forum of Mathematics, Pi
Volume	8
DOIs	https://doi.org/10.1017/fmp.2020.11
State	Published - Jan 5 2021

All Science Journal Classification (ASJC) codes

Analysis
Algebra and Number Theory
Statistics and Probability
Mathematical Physics
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1017/fmp.2020.11

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N2 - Let P1, . . ., Pm ∈ ℤ[y] be polynomials with distinct degrees, each having zero constant term. We show that any subset A of {1, . . ., N} with no nontrivial progressions of the form x, x + P1(y), . . ., x + Pm(y) has size |A| ≪ N/(log log N)cP1,...,Pm . Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

AB - Let P1, . . ., Pm ∈ ℤ[y] be polynomials with distinct degrees, each having zero constant term. We show that any subset A of {1, . . ., N} with no nontrivial progressions of the form x, x + P1(y), . . ., x + Pm(y) has size |A| ≪ N/(log log N)cP1,...,Pm . Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

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JO - Forum of Mathematics, Pi

JF - Forum of Mathematics, Pi

M1 - e16

ER -

Bounds for sets with no polynomial progressions

Abstract

All Science Journal Classification (ASJC) codes

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