Lattice points in algebraic cross-polytopes and simplices

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.