Multi-scale incidence geometry

Project Information

MUSING

Grant agreement ID: 101087499

DOI

10.3030/101087499

EC signature date 28 February 2023

Start date 1 August 2023

End date 31 July 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 362 842,50

EU contribution

€ 1 362 842,50

Coordinated by

JYVASKYLAN YLIOPISTO
Finland

Periodic Reporting for period 1 - MUSING (Multi-scale incidence geometry)

Reporting period: 2023-08-01 to 2026-01-31

Summary of the context and overall objectives of the project

The project Multi-scale incidence geometry lies at the interface of fractal geometry and incidence geometry. Fractal geometry studies sets with no smoothness. Examples in nature are ubiquitous, including trees, clouds, fractured coastlines, the Milky Way -- or the graph of Dow Jones. Fractal sets also arise from many other branches of mathematics, such as dynamical systems, stochastics, or harmonic analysis. A prototypical question in fractal geometry asks to calculate, or estimate, the "dimension" of a fractal set. As a classical example, the dimension of the coastline of Britain is roughly 1.25.

Incidence geometry is a branch of combinatorics. It asks questions of the following kind: given m < n, and a planar set containing n points, how many distinct lines can there exist which intersect the set in at least m points? A sharp answer to this question was already found by E. Szemerédi and W. Trotter in the 80s.

The question answered by Szemerédi and Trotter can be extended to a fractal geometric problem, as follows: given s < t, and a planar set of dimension t, how many lines can there exist that intersect the set in dimension at least s? This question is known as the Furstenberg set problem, and it was posed by the harmonic analyst T. Wolff in the late 90s.

While the Furstenberg set problem has formal similarity to the one solved by Szemerédi and Trotter, a key difference makes it harder: unlike cardinality, fractal dimension encodes information at many scales. Informally speaking, a clustered set of a 1000 points has lower dimension than a well-spread set with 1000 points. This difference is not seen by cardinality, but needs to be taken into account when solving the Furstenberg set problem, and related questions. The Furstenberg set problem is a question in multi-scale incidence geometry.

The project aims to solve, or make progress, in several well-known problems related to the Furstenberg set problem. Examples include estimating the dimension of projections and visible parts of fractal sets. The Furstenberg problem itself was on the original research agenda, but it was solved in 2023 by K. Ren, H. Wang, P. Shmerkin, and the PI around the same time when the project started. This was a leap forward in fractal geometry, and the field is currently witnessing a period of rapid development: one expects that the technology developed for the Furstenberg set problem will also help crack many related problems. One concrete example is the visibility problem, formulated around 2000: it asks to prove that if one looks at a planar fractal from a "random" direction, then the visible part has dimension at most one.

A broader question concerns the sharpness of the solution of the Furstenberg set problem. The 2023 solution is sharp in the lack of further hypotheses. However, fractals arising in real applications often exhibit some degree of statistical self-similarity, whereas known sharpness examples to the Furstenberg set theorem do not. It is therefore plausible that there are sharper versions of the Furstenberg set theorem, and related results in fractal geometry, under mild additional hypotheses. Besides solving the visibility problem, finding these minimal hypotheses is a focus of the project.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The field developed rapidly in the months before the start of the ERC project: the Furstenberg set problem, wide open at the time the proposal was composed in 2022, was fully solved in the month the project started. The PI in collaboration with P. Shmerkin first solved an "Ahlfors-regular" special case in June 2023. K. Ren and H. Wang completed the full solution in August 2023.

The speed of progress took the community by surprise, and meant that the ERC project started from a different position than the PI had anticipated in 2022. Instead of focusing on the Furstenberg set problem, the main task became to push even further, and to benefit from the fresh technology. Already in September 2023, the PI in collaboration with P. Shmerkin and N. de Saxcé used the new machinery to solve a Fourier-analytic problem posed by J. Bourgain in 2010.

The implications of the Furstenberg set theorem on the visibility problem remain under investigation. In 2024, the PI in collaboration with D. Dabrowski and M. Goering (a postdoc in the project) found a sharp answer to a closely related question on the size of "s-Nikodým sets". These are sets in the plane with the following peculiar property: for every point x in the set, there is an s-dimensional family of lines that only touch the set at x. We proved in 2024 that the dimension of an s-Nikodym set cannot exceed 2 - s. The techniques in the paper used ideas from the solution of the Furstenberg set problem.

Another main trend of the project asks: how sharp is the solution to the Furstenberg set problem? The solution in 2023 passed via the "Ahlfors-regular" special case. Roughly speaking, Ahlfors-regular sets enjoy the following statistical self-similarity property: if one picks a ball centred on the set and "zooms in", the part of the set inside the ball has the the same dimension as the whole set. Many "natural" fractals have this property (zooming into the Dow Jones graph still looks roughly like the Dow Jones graph), but not all sets do: a slightly artificial example would be the union of a ball and a point. While the resolution of the Furstenberg set conjecture is sharp in the class of general sets, it may be unsharp in the class of Ahlfors-regular sets.

One of the main achievements of the project has been to produce evidence supporting this hypothesis. To explain this, I need to mention that the Furstenberg set problem is related to the following projection problem: given a t-dimensional fractal set in the plane, and an s-dimensional set of directions, how large are the typical projections of the set in the given set of directions? A corollary of the Furstenberg set theorem yields a sharp answer to this question for general sets. But does the answer remain sharp if the set is assumed to be Ahlfors-regular? In 2024, the PI established a projection theorem for Ahlfors-regular sets which is much stronger than the sharp result for general sets. Despite the progress, plenty of work remains in this direction; notably, the possibility of improving the Furstenberg set theorem for Ahlfors-regular sets remains a key open question.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The state of the art in fractal geometry experienced a leap when the Furstenberg set problem was solved in 2023. In particular, many interesting problems which were still wide open in 2022 were suddenly left behind the state of the art in 2023. This claim should be understood at the highest level: the solutions of these problems are typically not direct corollaries of the Furstenberg set theorem. The problems simply became a big step more tractable in 2023.

Accordingly, the new results in the project can be divided into two categories (1) those applying the Furstenberg set theorem, more or less directly, to crack related problems, and (2) those pushing even further, either beyond the solution of the Furstenberg set problem, or in an entirely different direction. As of August 2025, the project has produced 5 result in category (1), including the solution to Bourgain's problem, mentioned in the previous section. The results in category (2) include the results on s-Nikodým sets, and projections of Ahlfors-regular sets, also discussed in the previous section.