dc.contributor.author | Orponen, Tuomas | |
dc.date.accessioned | 2023-01-05T07:14:59Z | |
dc.date.available | 2023-01-05T07:14:59Z | |
dc.date.issued | 2023 | |
dc.identifier.citation | Orponen, T. (2023). Additive properties of fractal sets on the parabola. <i>Annales Fennici mathematici</i>, <i>48</i>(1), 113-139. <a href="https://doi.org/10.54330/afm.125826" target="_blank">https://doi.org/10.54330/afm.125826</a> | |
dc.identifier.other | CONVID_164896905 | |
dc.identifier.uri | https://jyx.jyu.fi/handle/123456789/84774 | |
dc.description.abstract | Olkoon 0 ≤ s ≤ 1 ja P := {(t,t2) ∈ R2:t ∈ [−1,1]}. Jos K ⊂ P on suljettu ja dimHK = s, on suoraviivaista nähdä, että dimH(K + K) ≥ 2s. Paperin pääkorollaari kertoo, että jos 0 < s < 1, joukon K lisääminen vielä kerran kasvattaa summaa: dimH(K + K + K) ≥ 2s + ϵ, missä ϵ = ϵ(s) > 0. Väite päätellään seuraavasta L6-arviosta Frostman-mittojen Fourier-muunnoksille. Olkoon 0 < s <1 , ja olkoon μ on Borel-mitta joukossa P, joka toteuttaa ehdon μ(B(x,r)) ≤ rs kaikille x ∈ P ja r > 0. Silloin on olemassa ϵ = ϵ(s) > 0 ja R0 ≥ 1, joille seurava epäyhtälö pätee kaikille ‖μ^‖L6(B(R))6 ≤ R2−(2s+ϵ). Todistuksen keskeinen idea on muotoilla ongelma uudelleen sopivana δ-diskretoituna pisteiden ja ympyröiden välisenä insidenssiongelmana. Tämä geometrinen pulma palautuu lopulta (s, 2s)-Furstenberg-joukko-ongelmaan. | fi |
dc.description.abstract | Let 0 ≤ s ≤ 1, and let P := {(t,t2) ∈ R2:t ∈ [−1,1]}. If K ⊂ P is a closed set with dimHK = s, it is not hard to see that dimH(K + K) ≥ 2s. The main corollary of the paper states that if 0 0. This information is deduced from an L6 bound for the Fourier transforms of Frostman measures on P. If 0 < 1, and μ is a Borel measure on P satisfying μ(B(x,r)) ≤ rs for all x ∈ P and r > 0, then there exists ϵ = ϵ(s) > 0 such that ‖μ^‖L6(B(R))6 ≤ R2−(2s+ϵ) for all sufficiently large R ≥ 1. The proof is based on a reduction to a δ-discretised point-circle incidence problem, and eventually to the (s, 2s)-Furstenberg set problem.1, then adding K once more makes the sum slightly larger: dimH(K + K + K) ≥ 2s+ϵ, where ϵ = ϵ(s) > 0. This information is deduced from an L6 bound for the Fourier transforms of Frostman measures on P. If 0 < 1, and μ is a Borel measure on P satisfying μ(B(x,r)) ≤ rs for all x ∈ P and r > 0, then there exists ϵ = ϵ(s) > 0 such that ‖μ^‖L6(B(R))6 ≤ R2−(2s+ϵ) for all sufficiently large R ≥ 1. The proof is based on a reduction to a δ-discretised point-circle incidence problem, and eventually to the (s, 2s)-Furstenberg set problem. | en |
dc.format.mimetype | application/pdf | |
dc.language.iso | eng | |
dc.publisher | Finnish Mathematical Society | |
dc.relation.ispartofseries | Annales Fennici mathematici | |
dc.rights | CC BY-NC 4.0 | |
dc.subject.other | Fourier transforms | |
dc.subject.other | additive energies | |
dc.subject.other | Furstenberg sets | |
dc.subject.other | Frostman measures | |
dc.title | Additive properties of fractal sets on the parabola | |
dc.type | article | |
dc.identifier.urn | URN:NBN:fi:jyu-202301051129 | |
dc.contributor.laitos | Matematiikan ja tilastotieteen laitos | fi |
dc.contributor.laitos | Department of Mathematics and Statistics | en |
dc.type.uri | http://purl.org/eprint/type/JournalArticle | |
dc.type.coar | http://purl.org/coar/resource_type/c_2df8fbb1 | |
dc.description.reviewstatus | peerReviewed | |
dc.format.pagerange | 113-139 | |
dc.relation.issn | 2737-0690 | |
dc.relation.numberinseries | 1 | |
dc.relation.volume | 48 | |
dc.type.version | publishedVersion | |
dc.rights.copyright | © 2022 Annales Fennici Mathematici | |
dc.rights.accesslevel | openAccess | fi |
dc.subject.yso | Fourier'n sarjat | |
dc.format.content | fulltext | |
jyx.subject.uri | http://www.yso.fi/onto/yso/p8723 | |
dc.rights.url | https://creativecommons.org/licenses/by-nc/4.0/ | |
dc.relation.doi | 10.54330/afm.125826 | |
dc.type.okm | A1 | |