Nivat's conjecture holds for sums of two periodic configurations

Nivat's conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps $\mathbb Z^2 \rightarrow \mathcal A$ where $\mathcal A$ is a finite set of symbols. Such configurations are often understood as colorings of a two-dimensional square grid. Let $P_c(m,n)$ denote the number of distinct $m \times n$ block patterns occurring in a configuration $c$. Configurations satisfying $P_c(m,n) \leq mn$ for some $m,n \in \mathbb N$ are said to have low rectangular complexity. Nivat conjectured that such configurations are necessarily periodic. Recently, Kari and the author showed that low complexity configurations can be decomposed into a sum of periodic configurations. In this paper we show that if there are at most two components, Nivat's conjecture holds. As a corollary we obtain an alternative proof of a result of Cyr and Kra: If there exist $m,n \in \mathbb N$ such that $P_c(m,n) \leq mn/2$, then $c$ is periodic. The technique used in this paper combines the algebraic approach of Kari and the author with balanced sets of Cyr and Kra.Comment: Accepted for SOFSEM 2018. This version includes an appendix with proofs. 12 pages + references + appendi

An Algebraic Approach to Nivat's Conjecture

This thesis introduces a new, algebraic method to study multidimensional configurations, also sometimes called words, which have low pattern complexity. This is the setting of several open problems, most notably Nivat’s conjecture, which is a generalization of Morse-Hedlund theorem to two dimensions, and the periodic tiling problem by Lagarias and Wang. We represent configurations as formal power series over d variables where d is the dimension. This allows us to study the ideal of polynomial annihilators of the series. In the two-dimensional case we give a detailed description of the ideal, which can be applied to obtain partial results on the aforementioned combinatorial problems. In particular, we show that configurations of low complexity can be decomposed into sums of periodic configurations. In the two-dimensional case, one such decomposition can be described in terms of the annihilator ideal. We apply this knowledge to obtain the main result of this thesis – an asymptotic version of Nivat’s conjecture. We also prove Nivat’s conjecture for configurations which are sums of two periodic ones, and as a corollary reprove the main result of Cyr and Kra from [CK15].Algebrallinen lähestymistapa Nivat’n konjektuuriin Tässä väitöskirjassa esitetään uusi, algebrallinen lähestymistapa moniulotteisiin,matalan kompleksisuuden konfiguraatioihin. Näistä konfiguraatioista, joita moniulotteisiksi sanoiksikin kutsutaan, on esitetty useita avoimia ongelmia. Tärkeimpinä näistä ovat Nivat’n konjektuuri, joka on Morsen-Hedlundin lauseen kaksiulotteinen yleistys, sekä Lagariaksen ja Wangin jaksollinen tiilitysongelma. Väitöskirjan lähestymistavassa d-ulotteiset konfiguraatiot esitetään d:n muuttujan formaaleina potenssisarjoina. Tämä mahdollistaa konfiguraation polynomiannihilaattoreiden ihanteen tutkimisen. Väitöskirjassa selvitetään kaksiulotteisessa tapauksessa ihanteen rakenne tarkasti. Tätä hyödyntämällä saadaan uusia, osittaisia tuloksia koskien edellä mainittuja kombinatorisia ongelmia. Tarkemmin sanottuna väitöskirjassa todistetaan, että matalan kompleksisuuden konfiguraatiot voidaan hajottaa jaksollisten konfiguraatioiden summaksi. Kaksiulotteisessa tapauksessa eräs tällainen hajotelma saadaan annihilaattori-ihanteesta. Tämän avulla todistetaan asymptoottinen versio Nivat’n konjektuurista. Lisäksi osoitetaan Nivat’n konjektuuri oikeaksi konfiguraatioille, jotka ovat kahden jaksollisen konfiguraation summia, ja tämän seurauksena saadaan uusi todistus Cyrin ja Kran artikkelin [CK15] päätulokselle

Closed Choice and a Uniform Low Basis Theorem

We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, of the natural numbers, of Cantor space and of Baire space correspond to the class of computable functions, of functions computable with finitely many mind changes, of weakly computable functions and of effectively Borel measurable functions, respectively. We also prove that all these classes correspond to classes of non-deterministically computable functions with the respective spaces as advice spaces. Moreover, we prove that closed choice on Euclidean space can be considered as "locally compact choice" and it is obtained as product of closed choice on the natural numbers and on Cantor space. We also prove a Quotient Theorem for compact choice which shows that single-valued functions can be "divided" by compact choice in a certain sense. Another result is the Independent Choice Theorem, which provides a uniform proof that many choice principles are closed under composition. Finally, we also study the related class of low computable functions, which contains the class of weakly computable functions as well as the class of functions computable with finitely many mind changes. As one main result we prove a uniform version of the Low Basis Theorem that states that closed choice on Cantor space (and the Euclidean space) is low computable. We close with some related observations on the Turing jump operation and its initial topology

The Silent Hexagon: Explaining Comb Structures

The paper presents, and discusses, four candidate explanations of the structure, and construction, of the bees’ honeycomb. So far, philosophers have used one of these four explanations, based on the mathematical Honeycomb Conjecture, while the other three candidate explanations have been ignored. I use the four cases to resolve a dispute between Christopher Pincock (2012) and Alan Baker (2015) about the Honeycomb Conjecture explanation. Finally, I find that the two explanations focusing on the construction mechanism are more promising than those focusing exclusively on the resulting, optimal structure. The main reason for this is that optimal structures do not uniquely determine the relevant optimiza- tion leading to the optimal structure

Mass, Entropy and Holography in Asymptotically de Sitter Spaces

We propose a novel prescription for computing the boundary stress tensor and charges of asymptotically de Sitter (dS) spacetimes from data at early or late time infinity. If there is a holographic dual to dS spaces, defined analogously to the AdS/CFT correspondence, our methods compute the (Euclidean) stress tensor of the dual. We compute the masses of Schwarzschild-de Sitter black holes in four and five dimensions, and the masses and angular momenta of Kerr-de Sitter spaces in three dimensions. All these spaces are less massive than de Sitter, a fact which we use to qualitatively and quantitatively relate de Sitter entropy to the degeneracy of possible dual field theories. Our results in general dimension lead to a conjecture: Any asymptotically de Sitter spacetime with mass greater than de Sitter has a cosmological singularity. Finally, if a dual to de Sitter exists, the trace of our stress tensor computes the RG equation of the dual field theory. Cosmological time evolution corresponds to RG evolution in the dual. The RG evolution of the c function is then related to changes in accessible degrees of freedom in an expanding universe.Comment: 31 pages, LaTeX. v2: references and acknowledgements added, rewrite of "RG flow vs. cosmological evolution" section, log divergences commented on, typos corrected, comments on sign