A general nonlinear version of Roth's theorem on the real line

Abstract

Let N > 1 be a real number and ε > 0 be given. In this paper, we will prove that, for a measurable subset S of [0,N] with positive density ε , there must be patterns of the formwhere γ is convex and has some curvature constraints, t > δ (ε,γ)γ −1 (N) and δ (ε,γ) is a positive constant depending only on ε and γ , γ −1 is the inverse function of γ . Our result extends Bourgain's result [2] to the general curve γ . We use Bourgain's energy pigeonholing argument and Li's σ -uniformity argument.