Report on Current Developments in the Research Area
General Direction of the Field
The recent developments in the research area have been marked by significant advancements in the study of tiling problems, linear recurrence sequences, and exponential polynomials, as well as the generalization of percolation models. These advancements are pushing the boundaries of what is known and are contributing to the resolution of long-standing conjectures and open problems.
In the realm of tiling problems, there has been a notable shift towards proving the undecidability of translational tiling in higher dimensions. This direction is motivated by the desire to settle the conjecture that translational tiling of (\mathbb{Z}^n) with a monotile is undecidable for some fixed (n). Recent work has extended the undecidability results from 3-dimensional space to 4-dimensional space, using innovative techniques that lift tiling problems from lower to higher dimensions. This approach not only advances the field but also provides a clearer path towards resolving the broader conjecture.
The study of linear recurrence sequences (LRS) has seen a major breakthrough with the decidability of the Skolem Problem for algebraic LRS of order at most 4. This result completes the picture for the Skolem Problem in this context, addressing a question that has remained open for nearly a century. The decidability of the Skolem Problem for higher-order LRS is a significant milestone, as it opens up new avenues for research in the computational theory of LRS and their applications.
Exponential polynomials and their applications in identifying polygonal regions from Fourier samples have also seen innovative progress. The recent work has demonstrated that a small set of Fourier samples can uniquely determine certain classes of polygonal regions, with the size of the required sample set growing only logarithmically with the number of parameters. This result is particularly notable for its non-constructive methods, which offer a new perspective on the problem of identifying geometric shapes from limited data.
Percolation theory has been extended to a broader class of graphs, including adjacency graphs of rhombus tilings of the plane. This generalization includes aperiodic tilings like Penrose tilings, which were previously not considered in the context of percolation. The recent work shows that the entire graph of such tilings almost surely gets infected under a broader class of initial configurations, a result that extends the seminal work of van Enter on the infinite grid.
Noteworthy Papers
Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes: This paper significantly advances the undecidability conjecture in higher dimensions, using a novel lifting technique.
Completing the picture for the Skolem Problem on order-4 linear recurrence sequences: The decidability result for algebraic LRS of order 4 is a major breakthrough, closing a long-standing gap in the field.
Exponential polynomials and identification of polygonal regions from Fourier samples: The logarithmic growth in the sample size required to identify polygonal regions is a significant theoretical advancement.
Bootstrap percolation on rhombus tilings: The generalization of percolation results to aperiodic rhombus tilings is a notable extension of the field, with implications for broader graph structures.
