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Register a new user$SL_2(mathbb{Z})$-tilings of the torus, Coxeter-Conway friezes and Farey triangulations
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The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.
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We explain the notion of $q$-deformed real numbers introduced in our previous work and overview their main properties. We will also introduce $q$-deformed Conway-Coxeter friezes.
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group $PSL(2,mathbb{Z})$. These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of $PSL(2,mathbb{Z})$.
We study a $2 times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $mathrm{PSL}(2,mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.
We prove that $L(SL_2(textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(textbf{k}^2rtimes SL_2(textbf{k}))$ for $textbf{k}=mathbb{Q}$. Then we show how to modify the proof to handle $textbf{k}=mathbb{Z}$. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between $L(SL_2(textbf{k}))$ and $L^{infty}(Y)rtimes SL_2(textbf{k})$, where $SL_2(textbf{k})curvearrowright Y$ denotes the quotient of the algebraic action $SL_2(textbf{k})curvearrowright widehat{textbf{k}^2}$ by modding out the relation $phisim phi$, where $phi$, $phiin widehat{textbf{k}^2}$ and $phi(x, y):=phi(-x, -y)$ for all $(x, y)in textbf{k}^2$. As a by-product, we show $L(PSL_2(mathbb{Q}))$ is a maximal von Neumann subalgebra in $L^{infty}(Y)rtimes PSL_2(mathbb{Q})$; in particular, $PSL_2(mathbb{Q})curvearrowright Y$ is a prime action, i.e. it admits no non-trivial quotient actions.
We establish an uncertainty principle for functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ with constant support (where $p mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ for which $|text{supp}; {f}| = S$ must satisfy $|text{supp}; hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredis theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.
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