On Polynomial Progressions Inside Sets of Large Dimension

In this note we connect Sobolev estimates in the context of polynomial averages e.g. $$\| \int_0^1 \prod_{k=1}^m f_k(x-t^k) \|_{1} \leq \text{Const} \cdot 2^{-\text{const} \cdot l} \prod_{i=1}^m \| f_k \|_m$$ whenever some $f_i$ vanishes on $\{ |\xi| \leq 2^l \}$ to the existence of polynomial progressions inside of sets of sufficiently large Hausdorff dimension, in analogy with work of Peluse in the discrete context. Our strongest (unconditional) result builds off deep work of Hu-Lie and is as follows: suppose that $\mathcal{P} = \{P_1,P_2,P_3\}$ vanish at the origin at different rates, and that $E \subset [0,1]$ has sufficiently large Hausdorff dimension, $$1 - \text{const}(\mathcal{P}) < \text{dim}_H(E) < 1$$ and Hausdorff content bounded away from zero, sufficiently large in terms of its dimension. Then $E$ contains a non-trivial polynomial progression of the form $$\{ x , x - P_1(t), x - P_2(t), x - P_3(t) \} \subset E, \; \; \; t \neq 0.$$ We also provide a short proof that whenever $E$ has sufficiently large Hausdorff dimension and Fourier dimension $> 1/2$, it necessarily contains a non-trivial generalized three-term arithmetic progression of the form $$\{ x, x - \theta_1 t, x- \theta_2 t\} \subset E, \; \; \; \theta_i \in \mathbb{Q},\ t \neq 0.$$