Math Graduate Student Seminar

A central theme in Ramsey theory is the identification of structured patterns within large sets. A particularly celebrated result in this direction is Szemerédi's theorem, which asserts that any subset of the integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman later generalized this theorem to encompass more general polynomial progressions.

In this talk, we investigate an analogous problem for "sparse" subsets in the Euclidean setting. Specifically, let $\mathcal{P}= \{P_1, ......, P_k\}$ be a collection of polynomials with real coefficients, distinct degrees, and zero constant terms. We prove that any compact subset of the Torus with large Hausdorff dimension must contain the following nontrivial polynomial progression: $\{x,x+P_1(y), ......,x+P_k(y)\} $.

The proof relies on a specific formulation of the Sobolev smoothing inequality, inspired by Peluse's polynomial Szemerédi theorem in the finite field setting. As a consequence of this inequality, we establish that the divergence set for the pointwise convergence problem associated with the relevant polynomial multiple ergodic averages has Hausdorff dimension strictly less than one. This constitutes the first pointwise convergence result that extends beyond the classical almost everywhere convergence paradigm