Let $\bar{\mathcal{P}}$ denote the closed $n$-dimensional convex polytope subtended by the origin and the lattice points {$b\_{i} \textbf{e} \_ {i}$}, where {$\mathbf{e}\_{i}$} is the standard basis of $\mathbb{R}^{n}$. Define the Ehrhart function $L \_{\bar{\mathcal{P}}}(t) = | t \bar{\mathcal{P}} \cap \mathbb{Z} ^{n}|$, where $t \bar{ \mathcal{P} }$ denotes the $t$-dilate of $\bar{ \mathcal{P} }$. 


It is known that $L\_{\bar{ \mathcal{P} } }(t)$ is a polynomial of degree $n$ in $t \in \mathbb{N}$ if $\bar{\mathcal{P}}$ is integral, i.e., {$b\_{i}$} are positive integers. My question is about the meaning of $L\_{\bar{ \mathcal{P} } }(t)$ when $t$ is *rational*.


**Question**: Suppose I'd like to calculate the number of non-negative integer solutions of 
\begin{eqnarray}
\frac{x\_1}{b\_1} + \cdots + \frac{x\_n}{b\_n} \leq t
\end{eqnarray}
for some positive rational $t$. Is there a way to compute this number from the aforementioned Ehrhart polynomial (via some interpolation method, etc.) or is some other machinery necessary?


Thanks in advance!



Rational dilates of integral convex polytopes
R. Diaz, S. Robins, and I studied your question for the inequality $b\_1 x\_1 + \dots + b\_d x\_d \le t$ for integral $t$ (which gives a rational polytope) in


* *The Frobenius problem, rational polytopes, and Fourier-Dedekind Sums*, Journal of Number Theory 96 (2002), 1–21, doi:[10.1006/jnth.2002.2786](https://doi.org/10.1006/jnth.2002.2786), arXiv:[math.NT/0204035](http://arxiv.org/abs/math.NT/0204035).


The case where $t$ is truly a rational variable is more complicated. A starting point is


* Eva Linke, *Rational Ehrhart quasi-polynomials*, Journal of Combinatorial Theory, Series A **118** Issue 7 (2011) 1966-1978, doi:[10.1016/j.jcta.2011.03.007](https://doi.org/10.1016/j.jcta.2011.03.007), arXiv:[1006.5612](http://arxiv.org/abs/1006.5612)


