Work

Bounds in a popular multidimensional nonlinear Roth theorem

Year: 2024

Type: article

Abstract: A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form , , . We obtain a multidimensional version of this result, which can be regarded as a first step toward effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective “popular” version of this result, showing that every dense set has some non‐zero such that the number of configurations with difference parameter is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower‐type bounds encountered in the popular linear Roth theorem.

Source: Journal of the London Mathematical Society

DOI: 10.1112/jlms.70019

Authors: Sarah Peluse, Sean Prendiville, Xuancheng Shao

Institutions: Stanford University, Lancaster University, University of Kentucky

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Arithmetic Structure in Dense Sets, Topics in Analytic Number Theory and Additive Combinatorics