Polynomial Roth Theorems on Sets of Fractional Dimensions

Let E subset of R be a closed set of Hausdorff dimension alpha is an element of (0, 1). Let P : R -> 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 Laba 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].