activity
20242026
collaborators

12 papers

math.CA2026

Planar sets with large visible parts

Tuomas Orponen

I construct a compact subset of the plane whose visible parts are -dimensional in all directions. This disproves the visibility conjecture. The value c…

math.CO2026

Visible parts and lower bounds on point-ray incidences

Tuomas Orponen, Alex Rutar

Let be a compact set. For , let be the visible part of in direction . We prove that $\operatorname{di…

math.CA2026

Projections, Furstenberg sets, and the sum-product problem

Tuomas Orponen, Pablo Shmerkin

We make progress on two interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, an…

math.CA2026

Furstenberg-type estimates under mild non-concentration assumptions

Tuomas Orponen, Pablo Shmerkin

We prove sharp -discretised versions of some variants of the Furstenberg set problem under weaker or different non-concentration assumptions compared to previous works.

math.CA2026

Nikodým maximal function with restricted directions

Tuomas Orponen, Hrit Roy

We study the planar Nikodým maximal operator associated to a direction set . We show that the quasi-Assouad dimension $s := \dim_{\…

math.CA2025

Uniformly perfect measures on strictly convex planar graphs are -flattening

Amir Algom, Tuomas Orponen

Uniformly perfect measures are a common generalisation of Ahlfors regular measures, self-conformal measures on the line, and their push-forwards under sufficiently regular maps. We…