12 citations · 44 across the 14 of their papers we have counts for
11 papers · 1 filter
Arithmetic regularity as an alternative to transference
Sam Chow, Sean Prendiville, Santiago Vazquez
Since Green (2005), the Fourier-analytic transference principle has dominated the landscape of combinatorial theorems relative to sparse arithmetic sets. We demonstrate a different…
Bounds in a popular multidimensional nonlinear Roth theorem
Sarah Peluse, Sean Prendiville, Xuancheng Shao
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…
Counting monochromatic solutions to diagonal Diophantine equations
Sean Prendiville
We show how to adapt the Hardy--Littlewood circle method to count monochromatic solutions to diagonal Diophantine equations. This delivers a lower bound which is optimal up to abso…
A polylogarithmic bound in the nonlinear Roth theorem
Sarah Peluse, Sean Prendiville
We show that sets of integers lacking the configuration , , have at most polylogarithmic density.
Quantitative bounds in the nonlinear Roth theorem
Sarah Peluse, Sean Prendiville
We show that there exists such that any subset of of density at least contains a nontrivial progression of the form . This…
Rado's criterion over squares and higher powers
Sam Chow, Sofia Lindqvist, Sean Prendiville
We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equatio…
A transference approach to a Roth-type theorem in the squares
Tim Browning, Sean Prendiville
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…
Four variants of the Fourier-analytic transference principle
Sean Prendiville
We survey four instances of the Fourier analytic 'transference principle' or 'dense model lemma', which allows one to approximate an unbounded function on the integers by a bounded…
Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case
Sean Prendiville
We obtain quantitative bounds in the polynomial Szemerédi theorem of Bergelson and Leibman, provided the polynomials are homogeneous and of the same degree. Such configurations inc…
Improvements in Birch's theorem on forms in many variables
T. D. Browning, Sean Prendiville
We show that a non-singular integral form of degree d is soluble non-trivially over the integers if and only if it is soluble non-trivially over the reals and the p-adic numbers, p…
Near-optimal mean value estimates for multidimensional Weyl sums
Scott T. Parsell, Sean M. Prendiville, Trevor D. Wooley
We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the…