A transference approach to a Roth-type theorem in the squares
By Tim D Browning, Sean M. Prendiville. 2015
46 citations recorded by the source corpus.
We show that any subset of the squares of positive relative upper density contains non-trivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/396c12ee64a35459c22b29cf07eb6b3b77eb8f17
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
Citations in the corpus
- Higher order Fourier analysis of multiplicative functions and applications (2014, methodology): “lysis tools, and the circle method have been used successfully to prove partition regularity of equations that enjoy some linearity features (also for non-linear equations with at least five variables [8, 42, 43, 47, 48, 68]), we have not found such tools adequate for the fully non-linear setup we ar”