  The number of lattice points $\left| tP \cap \mathbb{Z}^d \right|$, as a
function of the real variable $t>1$ is studied, where $P \subset \mathbb{R}^d$
belongs to a special class of algebraic cross-polytopes and simplices. It is
shown that the number of lattice points can be approximated by an explicitly
given polynomial of $t$ depending only on $P$. The error term is related to a
simultaneous Diophantine approximation problem for algebraic numbers, as in
Schmidt's theorem. The main ingredients of the proof are a Poisson summation
formula for general algebraic polytopes, and a representation of the Fourier
transform of the characteristic function of an arbitrary simplex in the form of
a complex line integral.
