Polynomial Roth Theorems on Sets of Fractional Dimensions. (6th January 2021)

• Record Type: Journal Article
• Authors: Fraser, Robert; Guo, Shaoming; Pramanik, Malabika
• Is Part Of: International mathematics research notices. Volume 2022:Number 10(2022)
• Page Start: 7809
• Page End: 7838
• Publication Date: 2021-01-06
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa377
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: Let $E\subset{\mathbb{R}}$ be a closed set of Hausdorff dimension $\alpha \in (0, 1)$ . Let $P: {\mathbb{R}}\to{\mathbb{R}}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$ . Our result extends the one of Łaba and Pramanik [ 11 ] to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot et al. [ 7 ].