Remarks on dimension of unions of curves. (April 2023)

• Record Type: Journal Article
• Authors: Ham, Seheon; Ko, Hyerim; Lee, Sanghyuk; Oh, Sewook
• Is Part Of: Nonlinear analysis. Volume 229(2023)
• Publication Date: 2023-04
• Subjects: 42B25 -- 28A75; Union of curves -- Average over curves -- Local smoothing estimates; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113207
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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 γ in R d, i.e., x + t γ, ( x, t ) ∈ R d × ( 0, ∞ ) . For a Borel set F ⊂ R d × ( 0, ∞ ), we show the unions of curves ⋃ ( x, t ) ∈ F ( x + t γ ) has Hausdorff dimension at least α + 1 whenever F has Hausdorff dimension bigger than α, α ∈ ( 0, d − 1 ) . We also obtain results for unions of curves generated by multi-parameter dilation of γ . One of the main ingredients is a local smoothing type estimate (for averages over curves) relative to fractal measures.