I was reading that paper about some quantitative results on simplices in subsets of vector spaces over finite fields. I think that I understood most of the technical details of that paper. However, I did not understand how Theorem 3.1 implies Theorem 1.3.

Theorem 1.3. Let $E\subset \mathbb{F}_q^d, d>\tbinom{k+1}{2},$ such that $|E|\geq Cq^{\frac{k}{k+1}d+\frac{k}{2}}$ with a sufficiently large constant $C>0$. Then $E$ contains an isometric copy of every $k$-simplex.

Theorem 3.1. Let $E\subset \mathbb{F}_q^d, d>\tbinom{k+1}{2},$ such that $|E|\geq Cq^{\frac{k}{k+1}d+\frac{k}{2}}$ with a sufficiently large constant $C>0$. Then for every side length $l_k,\ l_k\in (\mathbb{F}_q^{\times})^{\tbinom{k+1}{2}}$ we have $|\mathcal{T}_{l_k}|>0.$ Furthermore, $|\mathcal{T}_{l_k}|\sim |E|^{k+1}q^{-\tbinom{k+1}{2}}.$

Using this theorem we recover the main result of the paper using the following linear algebraic observation.

Lemma 3.2. Let $P$ be a simplex with vertices $v_0,v_1,\dots v_k, \ v_j\in \mathbb{F}_q^d.$ Let $P'$ be another simplex with vertices $v'_0,v'_1,\dots,v'_k$. Suppose that $$\lVert v_i-v_j\rVert=\lVert v'_i-v'_j\rVert$$ for all $i,j$. Then there exists an orthogonal, affine transformation $O$ on $\mathbb{F}_q^d$ such that $O(P)=P'$.

On page 4 authors write that "This representation does not, in general, always embody a simplex, as $\mathcal{T}_{l_k}$ is not guaranteed to be in general position. However, as we show below, "legitimate" $k$-simplices are equivalent up to an orthogonal transformation."

Question 1: I am really confused how these observations imply Theorem 1.3.

Question 2: What does mean "isometric copy of $k$-simplex"?

I was reading Hart and Iosevich - Ubiquity of simplices in subsets of vector spaces over finite fields about some quantitative results on simplices in subsets of vector spaces over finite fields. I think that I understood most of the technical details of that paper. However, I did not understand how Theorem 3.1 implies Theorem 1.3.$\newcommand\card[1]{\lvert#1\rvert}\newcommand\norm[1]{\lVert#1\rVert}$

Theorem 1.3. Let $E\subset \mathbb{F}_q^d$, $d>\tbinom{k+1}{2}$ such that $\card E\geq Cq^{\frac{k}{k+1}d+\frac{k}{2}}$ with a sufficiently large constant $C>0$. Then $E$ contains an isometric copy of every $k$-simplex.

Theorem 3.1. Let $E\subset \mathbb{F}_q^d$, $d>\tbinom{k+1}{2}$, such that $\card E \geq Cq^{\frac{k}{k+1}d+\frac{k}{2}}$ with a sufficiently large constant $C>0$. Then for every side length $l_k$, $l_k\in (\mathbb{F}_q^{\times})^{\tbinom{k+1}{2}}$ we have $\card{\mathcal{T}_{l_k}}>0$. Furthermore, $\card{\mathcal{T}_{l_k}}\sim \card E^{k+1}q^{-\tbinom{k+1}{2}}$.

Using this theorem we recover the main result of the paper using the following linear algebraic observation.

Lemma 3.2. Let $P$ be a simplex with vertices $v_0,v_1,\dotsc, v_k$, $v_j\in \mathbb{F}_q^d$. Let $P'$ be another simplex with vertices $v'_0,v'_1,\dotsc,v'_k$. Suppose that $$\norm{v_i-v_j}=\norm{v'_i-v'_j}$$ for all $i,j$. Then there exists an orthogonal, affine transformation $O$ on $\mathbb{F}_q^d$ such that $O(P)=P'$.

On page 4 authors write that "This representation does not, in general, always embody a simplex, as $\mathcal{T}_{l_k}$ is not guaranteed to be in general position. However, as we show below, "legitimate" $k$-simplices are equivalent up to an orthogonal transformation."

Question 1: I am really confused how these observations imply Theorem 1.3.

Question 2: What does "isometric copy of a $k$-simplex" mean?