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# 数学代写|丢番图逼近代写Diophantine approximation代考|MAS7215 Parameterizations and special windows

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## 数学代写|丢番图逼近代写DIOPHANTINE APPROXIMATION代考|Parameterizations and special windows

Let $E$ be a $d$-dimensional subspace of $\mathbb{R}^k$, and let us assume that $E$ can be written as
$$E=\left{(x, L(x)): x \in \mathbb{R}^d\right},$$
where $L: \mathbb{R}^d \rightarrow \mathbb{R}^{k-d}$ is a linear function. This can always be achieved by a relabelling of the standard basis vectors, so for simplicity we will only work with subspaces $E$ which can be written this way. For each $1 \leq i \leq k-d$, we define the linear form $L_i: \mathbb{R}^d \rightarrow \mathbb{R}$ by
$$L_i(x)=L(x)i=\sum{j=1}^d \alpha_{i j} x_j,$$

and we use the points $\left{\alpha_{i j}\right} \in \mathbb{R}^{d(k-d)}$ to parametrize the choice of $E$.
As a reference point, when allowing $E$ to vary, we also make use of the fixed $(k-d)$-dimensional subspace $F_\rho$ of $\mathbb{R}^k$ defined by
$$F_\rho=\left{(0, \ldots, 0, y): y \in \mathbb{R}^{k-d}\right},$$
and we let $\rho: \mathbb{R}^k \rightarrow E$ and $\rho^*: \mathbb{R}^k \rightarrow F_\rho$ be the projections onto $E$ and $F_\rho$ with respect to the decomposition $\mathbb{R}^k=E+F_\rho$ (note that $E$ and $F_\rho$ are complementary subspaces of $\mathbb{R}^k$ ). Our notational use of $\pi$ and $\rho$ is intended to be suggestive of the fact that $F_\pi$ is the subspace which gives the projection defining $Y$ (hence the letter $\pi$ ), while $F_\rho$ is the subspace with which we reference $E$ (hence the letter $\rho$ ). We write $\mathcal{W}=\mathcal{S} \cap F_\rho$, and for convenience we also refer to this set, in addition to $\mathcal{W}_\pi$, as the window defining $Y$. This slight ambiguity should not cause any confusion in the arguments below.

## 数学代写|丢番图逼近代写DIOPHANTINE APPROXIMATION代考|Patches in cut and project sets

In analogy with the definition of ‘subword of length $n$ ‘ for a bi-infinite word, we now consider ‘patches of size $r$ ‘ in a cut and project set $Y$. It turns out that there is more than one reasonable choice for how to define patches of size $r$ in $Y$. We will work with two definitions, moving back and forth between them.

Assume that we are given a bounded convex set $\Omega \subseteq E$ which contains a neighborhood of 0 in $E$. For $y \in Y$ and $r \geq 0$ define $P_1(y, r)$, the type 1 patch of size $r$ at $y$, by
$$P_1(y, r):=\left{y^{\prime} \in Y: y^{\prime}-y \in r \Omega\right} .$$
Writing $\tilde{y}$ for the point in $\mathcal{S} \cap\left(\mathbb{Z}^k+s\right)$ with $\pi(\tilde{y})=y$, we define $P_2(y, r)$, the type 2 patch of size $r$ at $y$, by
$$P_2(y, r):=\left{y^{\prime} \in Y: \rho\left(\tilde{y^{\prime}}-\tilde{y}\right) \in r \Omega\right} .$$
Note that the point $\tilde{y}$ is uniquely determined by $y$ because of our standing assumption that $\left.\pi\right|_{\mathbb{Z}^k}$ is injective.

To rephrase the definitions, a type 1 patch consists of all points of $Y$ in a certain neighborhood of $y$ in $E$, while a type 2 patch consists of the projections of all points of $\mathcal{S} \cap \mathbb{Z}^k$ whose first $d$ coordinates are in a certain neighborhood of the first $d$ coordinates of $\tilde{y}$. Type 1 patches are more natural from the point of view of working within $E$, but the behavior of type 2 patches is more closely tied to the Diophantine properties of $L$.

## 数学代写|丟番图逼近代写DIOPHANTINE APPROXIMATION代考|Parameterizations and special windows

〈left 缺少或无法识别的分隔符

$$L_i(x)=L(x) i=\sum j=1^d \alpha_{i j} x_j,$$

\left 缺少或无法识别的分隔符

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