A polynomial Roth theorem on the real line

Abstract

For a polynomial P of degree greater than one, we show the existence of patterns of the form (x, x + t, x + P (t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain's approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.

On Maximal Functions With Curvature

“…Our first order of business is to establish Theorem 1.9 in the case when our set A has positive measure; this follows by specializing f = 1 A in the Proposition 6.1 below. This proposition is a multi-linear analogue of Bourgain's work [1] on non-linear Roth theorems and its subsequent extension [4]. Proposition 6.1.…”

“…Proof. Following the approach of [4], we minorize the convolution operators ϕ k * f with the conditional expectation operators…”

The non-resonant bilinear Hilbert--Carleson operator

“…Accordingly, the proof of the L p -boundedness of C d is based on some key new ideas: 1) developing a higher-order wave-packet analysis consistent with (29) that is compatible with the time-frequency approach to the classical Carleson operator C, and 2) introducing a local analysis adapted to the concepts of mass and counting function. 30 The n-dimensional version of Stein's conjecture on the polynomial Carleson operator was proved more recently -see [100] and [70] -by essentially combining the methodology in [77] with the n-dimensional Van der Corput estimates proved in [99].…”

“…The bilinear Hilbert transform. The generic formulation 31 of the main question within this direction can be stated as follows: 29 Here M j,a f (x) := e iax j f (x) represents a generalized modulation of order j ∈ N and parameter a ∈ R. 30 This latter aspect provides a novel tile discretization of the time-frequency plane with the consequence of eliminating certain exceptional sets from the analysis of C d . (These exceptional sets were present in all of the previously known approaches to the boundedness of the classical Carleson operator C = C 1 -see [17], [37] and [65]).…”

“…[39], [51]); b) from the number theory side via the so-called Ergodic Roth type theorem(s) (see e.g. [8], [9], [30], [58]); and c) from the harmonic analysis side as a natural curved analogue of the standard bilinear Hilbert transform considered above. 32 There is still an open problem regarding the maximal range for p, q, r -specifically the interval 1 2 < r ≤ 2 3 -that guarantees the boundedness of Ba.…”