Large Sets with Fourier Decay avoiding Patterns

Abstract: We discuss the construction of sets with large Fourier dimension avoiding certain families of linear and non-linear patterns. In other words, we construct sets which do not contain a certain subset of points arranged in a particular configuration, while also supporting probability measures whose Fourier transforms exhibit polynomial decay. Our analysis involves a discussion of the concentration of measure phenomenon in probability, and some oscillatory integral estimates. As particular applications of these methods, we will construct large sets of $\mathbf{T}^d$ not containing points $x_1,\dots,x_n$ solving linear equations of the form $a_1x_1 + ... a_n x_n = b$, and large subsets of planar curves with non-vanishing curvature which do not contain three points forming an isosceles triangle.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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