Polynomial Roth Theorems on Sets of Fractional Dimensions

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].