Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; (\mathbb{R}^{+})^{n})$ by the non-negative points $(x_{1}, \dots, x_{n}) \subset (\mathbb{R}^{+})^{n}$ which satisfy the inequality \begin{eqnarray} \omega_{1} x_{1} + \cdots + \omega_{n} x_{n} \leq 1. \end{eqnarray} Let $X$ be a non-trivial subset of the integers $\mathbb{Z}^{n}$. Define $\Delta(\Omega, X) = X \cap \Delta(\Omega, (\mathbb{R}^{+})^{n})$. It is well-known that \begin{eqnarray} |\Delta(\Omega, \mathbb{N}^{n})| \leq \frac{1}{n!} \prod_{i = 1}^{n} \frac{1}{\omega_{i}} \quad \text{and} \quad |\Delta(\Omega, (\mathbb{Z}^{+})^{n})| \leq \frac{1}{n!} \left(1 + \sum_{i = 1}^{n} \omega_{i} \right)^{n} \prod_{i = 1}^{n} \frac{1}{\omega_{i}}, \end{eqnarray} where $\mathbb{Z}^{+}$ denotes the set of non-negative integers.
Question(s): For the given bounds above, are any sharper bounds known? Given the similarity in form, are there formulas for other $X$ sets, say for integers greater than some arbitrary integer $c$ or integers satisfying some congruence condition (e.g., $a \equiv b$ mod $d$)?
(Update) The theory of Ehrhart polynomials is relevant to the question above.
Question: Suppose I'd like to use the Ehrhart machinery to count the number of non-negative integer solutions of $a_{1} x_{1} + \cdots + a_{n} x_{n} \leq r$ for a non-negative integer $r$ and positive integers {$a_{i}$}. How does one proceed?
Thanks!