Mathematics is also making progress by using new computational tools to solve problems that have remained intractable for decades. One example of this is the work in which Elvira Mayordomo, a researcher at the Aragón Engineering Research Institute (I3A) at the University of Zaragoza, is involved; she has just received the award for ‘Best Scientific Paper’ at the International Symposium on the Mathematical Foundations of Computer Science (Mathematical Foundations of Computer Science, MFCS), which has just taken place in Paris.
The study shows that both the distances between points and their projections retain a significant portion of the original information, provided that the elements involved are sufficiently independent of one another.
The article presents a new technique for measuring the information contained in geometric objects using what is known as Kolmogorov complexity, a theory that makes it possible to quantify how much information is required to describe an object. Using this approach, the researchers demonstrate that, even when a point is transformed into a distance or a projection onto a line, it retains a significant portion of the information that characterises it.
Its implications are significant for mathematical research. The new methodology enables progress to be made on open problems relating to fractal geometry, a field that studies extremely complex structures, similar to those found in natural phenomena such as coastlines, the branches of a tree or networks of blood vessels.
Specifically, the work improves upon some of the best known results concerning the so-called Falconer conjecture, one of the major open problems in modern geometry. This conjecture seeks to understand what happens to all the distances that can be obtained between points in a highly irregular set. The study also extends classical results on geometric projections developed by the mathematician Jean Bourgain.
The authors themselves point out that the technique is not only useful for solving the two problems studied in the article, but also constitutes a new methodology that is likely to have applications in other areas of geometry and computer science. In fact, it has already begun to be used in subsequent research into other mathematical problems related to Kakeya’s conjecture.
Key points of the article
- Develop a new mathematical tool for measuring the information contained in geometric objects using Kolmogorov complexity.
- It shows that distances and projections retain at least half the information of the original point under certain conditions.
- It improves on previous results relating to Falconer’s conjecture on sets of distances and generalises a well-known result by Bourgain on orthogonal projections.
- It opens up a new avenue of research, as the technique can be applied to other unsolved problems in fractal geometry and information theory.
The study brings together researchers from various international institutions and counts Elvira Mayordomo, from the COSMOS (Computer Science for Complex System Modelling) research group at I3A Unizar, amongst its authors. Her research focuses on computational theory, algorithmic complexity and their links to mathematics.
Access to the article:
Algorithmic Information Bounds for Distances and Orthogonal Projections. Peter Cholak, Marianna Csörnyei, Neil Lutz, Patrick Lutz, Elvira Mayordomo, D. M. Stull. https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.13
