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

Local Complexity of Delone Sets and Crystallinity

This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to X being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in this bound is best possible in all dimensions. For M(T), either M(T) is bounded or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package

Multidimensional extension of the Morse--Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of \ZZ^d definable by a first order formula in the Presburger arithmetic . With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of $\ZZ^d$ definable in in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often

Repetitive Delone Sets and Quasicrystals

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set whose patch-counting function N(T), for radius T, is finite for all T is called repetitive if there is a function M(T) such that every ball of radius M(T)+T contains a copy of each kind of patch of radius T that occurs in the set. This is equivalent to the minimality of an associated topological dynamical system with R^n-action. There is a lower bound for M(T) in terms of N(T), namely N(T) = O(M(T)^n), but no general upper bound. The complexity of a repetitive Delone set can be measured by the growth rate of its repetitivity function M(T). For example, M(T) is bounded if and only if the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and densely repetitive if M(T) = O(N(T))^{1/n}). We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive. In the reverse direction, we construct a repetitive Delone set in R^n which has M(T) = O(T(log T)^{2/n}(log log log T)^{4/n}), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets, and we propose considering them as a notion of "perfectly ordered quasicrystal".Comment: To appear in "Ergodic Theory and Dynamical Systems" vol.23 (2003). 37 pages. Uses packages latexsym, ifthen, cite and files amssym.def, amssym.te

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