A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
How much can heavy lines cover? (2024)


Dąbrowski, D., Orponen, T., & Wang, H. (2024). How much can heavy lines cover?. Journal of the London Mathematical Society, 109(5), Article e12910. https://doi.org/10.1112/jlms.12910


JYU-tekijät tai -toimittajat


Julkaisun tiedot

Julkaisun kaikki tekijät tai toimittajat: Dąbrowski, Damian; Orponen, Tuomas; Wang, Hong

Lehti tai sarja: Journal of the London Mathematical Society

ISSN: 0024-6107

eISSN: 1469-7750

Julkaisuvuosi: 2024

Ilmestymispäivä: 30.04.2024

Volyymi: 109

Lehden numero: 5

Artikkelinumero: e12910

Kustantaja: Wiley-Blackwell

Julkaisumaa: Britannia

Julkaisun kieli: englanti

DOI: https://doi.org/10.1112/jlms.12910

Julkaisun avoin saatavuus: Avoimesti saatavilla

Julkaisukanavan avoin saatavuus: Osittain avoin julkaisukanava

Julkaisu on rinnakkaistallennettu (JYX): https://jyx.jyu.fi/handle/123456789/94677

Rinnakkaistallenteen verkko-osoite (pre-print): https://arxiv.org/abs/2310.11219v1


Tiivistelmä

One formulation of Marstrand’s slicing theorem is the following. Assume that𝑡∈(1,2],and𝐵⊂ℝ2is a Borel set with 𝑡(𝐵)<∞. Then, for almost all directions𝑒∈𝑆1, 𝑡 almost all of 𝐵 is covered by lines𝓁parallel to 𝑒 with dim H(𝐵∩𝓁)=𝑡−1. We investigate the prospects of sharpening Marstrand’s result in the following sense: in a generic direction𝑒∈𝑆1, is it true that a strictly less than 𝑡-dimensional part of𝐵is covered by the heavy lines𝓁⊂ℝ2, namely those with dim H(𝐵∩𝓁)>𝑡−1? A positive answer for𝑡-regular sets𝐵⊂ℝ2was previously obtained by the first author. The answer for general Borel sets turns out to be negative for𝑡∈(1,32] and positive for𝑡∈(32,2]. More precisely, the heavy lines can cover up to amin{𝑡,3−𝑡} dimensional part of𝐵in a generic direction. We also consider the part of𝐵covered by the𝑠-heavy lines, namely those with dim H(𝐵∩𝓁)⩾𝑠for𝑠>𝑡−1. We establish a sharp answer to the question: how much can the𝑠-heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author’s previous result on Ahlfors-regular sets to the class of sub uniformly distributed sets.


YSO-asiasanat: mittateoria; fraktaalit


Liittyvät organisaatiot


Hankkeet, joissa julkaisu on tehty


OKM-raportointi: Kyllä

Raportointivuosi: 2024

Alustava JUFO-taso: 3


Viimeisin päivitys 2024-13-05 klo 18:25