On the torus, Hausdorff dimension above a threshold 1-epsilon forces arbitrary-length polynomial progressions for distinct-degree polynomials with zero constant terms.
A quadratic Roth theorem for sets with large Hausdorff dimensions
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abstract
Many results in harmonic analysis and geometric measure theory ensure the existence of geometric configurations under the largeness of sets, which are sometimes specified via the ball condition and Fourier decay. Recently, Kuca-Orponen-Sahlsten and Bruce-Pramanik proved Sarkozy-like theorems, which remove the Fourier decay condition and show that sets with large Hausdorff dimensions contain two-point patterns. This paper explores the existence of a three-point configuration that relies solely on the Hausdorff dimension.
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Polynomial Szemer\'edi for sets with large Hausdorff dimension on the Torus
On the torus, Hausdorff dimension above a threshold 1-epsilon forces arbitrary-length polynomial progressions for distinct-degree polynomials with zero constant terms.