Lisbon Webinar in Analysis and Differential Equations

Remarks on dimension of union of curves

Sala P3.10, Instituto Superior Técnico (com transmissão via Zoom)

Por Seheon Ham (Seoul National University, Republic of Korea).

We study an analogue of Marstrand's circle packing problem for curves in higher dimensions.

We consider collections of curves which are generated by translation and dilation of a curve $\gamma$ in $\mathbb R^d$, i.e., $ x + t \gamma$, $(x,t) \in \mathbb R^d \times (0,\infty)$.

For a Borel set $F \subset \mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in F} ( x+t\gamma )$ has Hausdorff dimension at least $\alpha+1$ whenever $F$ has Hausdorff dimension bigger than $\alpha\in (0, d-1)$.

We also obtain results for unions of curves generated by multi-parameter dilation of $\gamma$.

One of the main ingredients is a local smoothing type estimate (for averages over curves) relative to fractal measures.

This talk is bases on recent work with Herym Ko, Sanghyuk Lee, and Sewook Oh.

Transmissão via Zoom (pw: lisbonwade).

The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.