A set of positive integers $d\_1, \dots, d\_n$ describe a n-dimensional tetrahedron $T$ with the vertices
$$ (0,\dots,0), (1/d\_1,0,\dots,0), (0, 1/d\_2,\dots,0), \dots, (0,\dots,1/d\_n).$$
Let $L\_T(t)$ be the Ehrhart (quasi-)polynomial of $T$, i.e. the number of integer lattice points in $tT$.


I understand that typically everything beyond the first, second and last coefficients of $L\_T(t)$ are rather difficult to determine in general. Is there an approximation possible for the other coefficients? As this for a practical problem, I'm not so much interested in an upper and lower bound, but rather a good "guess".



Approximating Ehrhart Polynomial of Rational n-Tetrahedron
Actually, $L\_T(t)$ is *not* a polynomial (it's an honest quasipolynomial except for very special cases of $d\_1, \dots, d\_n$) and already the second "coefficient" is nontrivial, as is the last "coefficient". (The leading coefficient is a constant--the volume of T.) I'm not sure what exactly you're after in terms of approximating this, but here is one fact I'm aware of: let's write the quasipolynomial as $L\_T(t) = c\_{n-1}(t) \, t^{n-1} + \dots + c\_0(t)$; then each $c\_j(t)$ is periodic and so we can ask for its average value; let's call that $a\_j$. Thus $a\_{n-1} \, t^{n-1} + \dots + a\_0$ is a polynomial that somehow encodes how $L\_T(t)$ behaves "on average." It turns out this polynomial is a close relative of the Bernoulli polynomials (it's essentially what's known as a *Bernoulli-Barnes polynomial*); see, e.g., <http://math.sfsu.edu/beck/ccd.html>, p. 151.


