A lot of Minkowski related examples might work here. For example:

Let $D$ is the Discriminant of an algebraic number field $K$ with degree $n$ over $ \mathbb{Q}$, where $r_1$ and $r_2$ are the numbers of real and complex embeddings Then every class in the ideal class group of $K$ contains an integral ideal of norm at most $$\sqrt{|D|} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n} \ . $$

Understanding how this behaves for various fields sometimes uses that $\pi <4$. Other estimates connect this bound to Stirling's formula for $n!$.

Another related direction, also coming from Minkowski type arguments is the Minkowski-style proof of Lagrange's four square theorem, where $\pi >2$ is used. More accurate estimates might be used for results about other similar quadratic forms.

A lot of Minkowski related examples might work here. For example:

Let $D$ be the Discriminant of an algebraic number field $K$ with degree $n$ over $ \mathbb{Q}$, where $r_1$ and $r_2$ are the numbers of real and complex embeddings. Then every class in the ideal class group of $K$ contains an integral ideal of norm at most $$\sqrt{|D|} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n} \ . $$

Understanding how this behaves for various fields sometimes uses that $\pi <4$. Other estimates connect this bound to Stirling's formula for $n!$.

Another related direction, also coming from Minkowski type arguments is the Minkowski-style proof of Lagrange's four square theorem, where $\pi >2$ is used. More accurate estimates might be used for results about other similar quadratic forms.

A lot of Minkowski related examples might work here. For example:

Let $D$ is the Discriminant of an algebraic number field $K$ with degree $n$ over $ \mathbb{Q}$, where $r_1$ and $r_2$ are the numbers of real and complex embeddings Then every class in the ideal class group of $K$ contains an integral ideal of norm not at most $$\sqrt{|D|} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n} \ . $$

Understanding how this behaves for various fields sometimes uses that $\pi <4$. Other estimates connect this bound to Stirling's formula for $n!$.

Another related direction, also coming from Minkowski type arguments is the Minkowski-style proof of Lagrange's four square theorem, where $\pi >2$ is used.

A lot of Minkowski related examples might work here. For example:

Let $D$ is the Discriminant of an algebraic number field $K$ with degree $n$ over $ \mathbb{Q}$, where $r_1$ and $r_2$ are the numbers of real and complex embeddings Then every class in the ideal class group of $K$ contains an integral ideal of norm at most $$\sqrt{|D|} \left(\frac{4}{\pi}\right)^{r_2} \frac{n!}{n^n} \ . $$

Understanding how this behaves for various fields sometimes uses that $\pi <4$. Other estimates connect this bound to Stirling's formula for $n!$.

Another related direction, also coming from Minkowski type arguments is the Minkowski-style proof of Lagrange's four square theorem, where $\pi >2$ is used. More accurate estimates might be used for results about other similar quadratic forms.