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Question 1: In the statement of Theorem 3.1, $T_{l_k}$ is [edit in response to comments] an isometric copy of a $k$-simplex in the set $E$; the conclusion $\lvert T_{l_k}\rvert > 0$ is precisely the conclusion that $T_{\ell_k}$ is not an isometric copy of a trivial (empty) simplex. Note that this is all that is claimed in Theorem 1.3 — while the title of the paper talks about simplices, the main theorem only claims isometric copies of them, which could fail to be in general position.

Question 2: At the beginning of section 3 (page 4), the authors define simplices in terms of a certain “distance” $\lVert \cdot \rVert$. One simplex $T$ is an isometric copy of another, $T'$, if $T'$ can be obtained from $T$ by a transformation $O$ that preserves distances between points within the simplex, i.e. for $x_i$, $x_j$ points of $T$ we have $\lVert O(x_i) - O(x_j) \rVert = \lVert x_i - x_j \rVert$. You can see in Lemma 3.2 that such a transformation can always be taken to be an orthogonal transformation of $\mathbb{F}_q^d$.

Not every distance set $l_k$ will give you a genuine $k$-simplex. But the theorem gives you a realization in $E$ of every set of distances $l_k$, including the ones coming from genuine $k$-simplices. There are a bunch of $T_{l_k}$'s, some of which are $k$-simplices, and they're all nonempty (when the conditions of the theorem are satisfied).

[Complete rewrite in response to comments, having realized where my previous misunderstanding was.]

Question 1: In the statement of Theorem 3.1, $T_{l_k}$ is [edit in response to comments] an isometric copy of a $k$-simplex in the set $E$; the conclusion $|T_{l_k}| > 0$ is precisely the conclusion that $T_{\ell_k}$ is not an isometric copy of a trivial (empty) simplex. Note that this is all that is claimed in Theorem 1.3 -- while the title of the paper talks about simplices, the main theorem only claims isometric copies of them, which could fail to be in general position.

Question 2: At the beginning of section 3 (page 4), the authors define simplices in terms of a certain ``distance'' $\| \cdot \|$. One simplex $T$ is an isometric copy of another, $T'$, if $T'$ can be obtained from $T$ by a transformation $O$ that preserves distances between points within the simplex, i.e. for $x_i, x_j$ points of $T$ we have $\| O(x_i) - O(x_i) \| = \| x_i - x_j \|$. You can see in Lemma 3.2 that such a transformation can always be taken to be an orthogonal transformation of $\mathbb{F}_q^d$.

Question 1: In the statement of Theorem 3.1, $T_{l_k}$ is [edit in response to comments] an isometric copy of a $k$-simplex in the set $E$; the conclusion $\lvert T_{l_k}\rvert > 0$ is precisely the conclusion that $T_{\ell_k}$ is not an isometric copy of a trivial (empty) simplex. Note that this is all that is claimed in Theorem 1.3 — while the title of the paper talks about simplices, the main theorem only claims isometric copies of them, which could fail to be in general position.

Question 2: At the beginning of section 3 (page 4), the authors define simplices in terms of a certain “distance” $\lVert \cdot \rVert$. One simplex $T$ is an isometric copy of another, $T'$, if $T'$ can be obtained from $T$ by a transformation $O$ that preserves distances between points within the simplex, i.e. for $x_i$, $x_j$ points of $T$ we have $\lVert O(x_i) - O(x_j) \rVert = \lVert x_i - x_j \rVert$. You can see in Lemma 3.2 that such a transformation can always be taken to be an orthogonal transformation of $\mathbb{F}_q^d$.

Question 1: In the statement of Theorem 3.1, $T_{l_k}$ is a $k$-simplex in the set $E$; the conclusion $|T_{l_k}| > 0$ is precisely the conclusion that this simplex is not the trivial (empty) one.

Question 2: At the beginning of section 3 (page 4), the authors define simplices in terms of a certain ``distance'' $\| \cdot \|$. One simplex $T$ is an isometric copy of another, $T'$, if $T'$ can be obtained from $T$ by a transformation $O$ that preserves distances between points within the simplex, i.e. for $x_i, x_j$ points of $T$ we have $\| O(x_i) - O(x_i) \| = \| x_i - x_j \|$. You can see in Lemma 3.2 that such a transformation can always be taken to be an orthogonal transformation of $\mathbb{F}_q^d$.

Question 1: In the statement of Theorem 3.1, $T_{l_k}$ is [edit in response to comments] an isometric copy of a $k$-simplex in the set $E$; the conclusion $|T_{l_k}| > 0$ is precisely the conclusion that $T_{\ell_k}$ is not an isometric copy of a trivial (empty) simplex. Note that this is all that is claimed in Theorem 1.3 -- while the title of the paper talks about simplices, the main theorem only claims isometric copies of them, which could fail to be in general position.

Question 2: At the beginning of section 3 (page 4), the authors define simplices in terms of a certain ``distance'' $\| \cdot \|$. One simplex $T$ is an isometric copy of another, $T'$, if $T'$ can be obtained from $T$ by a transformation $O$ that preserves distances between points within the simplex, i.e. for $x_i, x_j$ points of $T$ we have $\| O(x_i) - O(x_i) \| = \| x_i - x_j \|$. You can see in Lemma 3.2 that such a transformation can always be taken to be an orthogonal transformation of $\mathbb{F}_q^d$.