A Transference Approach to a Roth-Type Theorem in the Squares

Abstract

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.

Higher order Fourier analysis of multiplicative functions and applications

“…Although combinatorial tools, Fourier analysis 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 are interested in. Instead, our main tool is going to be the structural result of Theorem 2.1.…”

“…The group G is endowed with the corresponding geodesic distance, which we denote by d G . This distance is invariant under right translations 8 .…”

Roth–Waring–Goldbach

“…We now verify this inequality by following the corresponding argument in the proof of [10,Lemma 6.3]. Let a n ∈ C be such that |a n | 1 and ψ(n) = a n µ(n), for n ∈ [N ].…”

“…(see the discussion surrounding [10,Lemma 6.3]). In our quest to establish (5.3), we begin by noting that if δ L −2v/ε 0 then by (5.2) we have…”

“…Using this mechanism, he showed that any subset of the primes with positive relative density contains nontrivial three-term arithmetic progressions. Recently Browning and Prendiville [10] have shown Green's transference method to be versatile, establishing a theorem of Roth-Waring type for five squares. They were able to transfer results from the integers to the squares, thereby obtaining a Roth- Then |A| ≪ c,k,ε X log X (log log log log X)…”