A polynomial Roth theorem on the real line. (English) Zbl 1432.42005
For a polynomial \(P:\mathbb{R}\to\mathbb{R}\), denote by \(\|P\|\) the \(\ell^1\) sum of the coefficients of \(P\). Let \(M,\,N\ge1\) be two real numbers. Assume that \(\varepsilon>0\) and \(S\) is a measurable subset of \([0,N]\) satisfying \(|S|\ge\varepsilon N\). Let \(P:\mathbb{R}\to\mathbb{R}\) be a monic polynomial of degree \(d>1\) without constant term that satisfies \(\|P\|\le M\).
In this article, the authors proved that there exists a positive constant \(\delta:=\delta(\varepsilon,M,d)\) such that one can find that \[x,x+t,x+P(t)\in S, \] with \(t>\delta N^{1/d}\), satisfies \((\log\log\delta^{-1})^{-\frac16}\gtrsim\varepsilon\). This is an extension of a result obtained by J. Bourgain [J. Anal. Math. 50, 169–181 (1988; Zbl 0675.42010)]. The proof for the above result is based on a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.
In this article, the authors proved that there exists a positive constant \(\delta:=\delta(\varepsilon,M,d)\) such that one can find that \[x,x+t,x+P(t)\in S, \] with \(t>\delta N^{1/d}\), satisfies \((\log\log\delta^{-1})^{-\frac16}\gtrsim\varepsilon\). This is an extension of a result obtained by J. Bourgain [J. Anal. Math. 50, 169–181 (1988; Zbl 0675.42010)]. The proof for the above result is based on a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.
Reviewer: Sibei Yang (Lanzhou)
MSC:
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
Keywords:
polynomial Roth theoremCitations:
Zbl 0675.42010References:
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